From f7150ffe1cd8466ce4943e99ba07562a79aaf5b4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Fri, 30 Jun 2023 14:54:34 +0200 Subject: [PATCH] added fluid mechanics and updated website --- .github/workflows/buildpdf.yml | 8 +- .../4th/Fluid_mechanics/Fluid_mechanics.tex | 244 +++++- index.html | 36 +- main_math.idx | 714 +++++++++--------- main_math.ilg | 6 +- main_math.ind | 655 ++++++++-------- main_physics.idx | 462 ++++++++++++ main_physics.ilg | 6 + main_physics.ind | 520 +++++++++++++ preamble_formulas.sty | 2 +- 10 files changed, 1950 insertions(+), 703 deletions(-) create mode 100644 main_physics.idx create mode 100644 main_physics.ilg create mode 100644 main_physics.ind diff --git a/.github/workflows/buildpdf.yml b/.github/workflows/buildpdf.yml index 9df76bf..d609b30 100644 --- a/.github/workflows/buildpdf.yml +++ b/.github/workflows/buildpdf.yml @@ -190,7 +190,12 @@ jobs: with: root_file: Quantum_physics.tex working_directory: Physics/3rd/Quantum_physics/ - + # PHYSICS 4 + - name: Compile - QF + uses: xu-cheng/latex-action@v2 + with: + root_file: Fluid_mechanics.tex + working_directory: Physics/4th/Fluid_mechanics/ - name: Create Release id: create_release uses: actions/create-release@v1 @@ -239,4 +244,5 @@ jobs: Physics/1st/Mechanics_and_special_relativity/Mechanics_and_special_relativity.pdf Physics/2nd/Structure_of_matter_and_thermodynamics/Structure_of_matter_and_thermodynamics.pdf Physics/3rd/Quantum_physics/Quantum_physics.pdf + Physics/4th/Fluid_mechanics/Fluid_mechanics.pdf github-token: ${{ secrets.GITHUB_TOKEN }} diff --git a/Physics/4th/Fluid_mechanics/Fluid_mechanics.tex b/Physics/4th/Fluid_mechanics/Fluid_mechanics.tex index 9f8270f..565d718 100644 --- a/Physics/4th/Fluid_mechanics/Fluid_mechanics.tex +++ b/Physics/4th/Fluid_mechanics/Fluid_mechanics.tex @@ -5,8 +5,8 @@ \begin{multicols}{2}[\section{Fluid mechanics}] \subsection{Equations of motion} \subsubsection{Euler's equations} - In this section we will describe the motion of a fluid with a set of equation that result from the conservation of mass, momentum and energy. From what follows, let $D\subseteq \RR^3$ be a region filled with a fluid. For each time $t$ and $\vf{x}\in D$ we assume that the fluid has a well-defined mass density $\rho(t,\vf{x})$\footnote{The assumption that $\rho$ exists is a continuum assumption. Clearly, it does not hold if the molecular structure of matter is taken into account. For most macroscopic phenomena occurring in nature, it is believed that this assumption is extremely accurate.}. Finally, we denoted by $\vf{u}(t,\vf{x})$ the velocity of the fluid at time $t$ and position $\vf{x}$. For the moment, we will also assume that $\rho$ and $\vf{u}$ are smooth functions. - \begin{proposition}[Conservation of mass] + In this section we will describe the motion of a fluid with a set of equation that result from the conservation of mass, momentum and energy. From what follows, let $D\subseteq \RR^3$ be a region filled with a fluid. For each time $t$ and $\vf{x}\in D$ we assume that the fluid has a well-defined mass density $\rho(\vf{x},t)$\footnote{The assumption that $\rho$ exists is a continuum assumption. Clearly, it does not hold if the molecular structure of matter is taken into account. For most macroscopic phenomena occurring in nature, it is believed that this assumption is extremely accurate.}. Finally, we denoted by $\vf{u}(\vf{x},t)$ the velocity of the fluid at time $t$ and position $\vf{x}$. For the moment, we will also assume that $\rho$ and $\vf{u}$ are smooth functions. + \begin{proposition}[Conservation of mass]\label{FLM:conservationofmass} Let $W\subseteq D$ be a fixed subregion of $D$. Then: $$ \dv{}{t}\int_W\rho\dd{V}=-\int_{\Fr{W}}\rho\vf{u}\cdot\dd{\vf{S}} @@ -40,7 +40,7 @@ \end{sproof} For any continuum, forces acting on a piece of material are of two types. First, there are forces of stress, whereby the piece of material is acted on by forces across its surface by the rest of the continuum. Second, there are external, or body, forces such as gravity or a magnetic field, which exert a force per unit volume on the continuum. \begin{definition}[Ideal fluid] - An \emph{ideal fluid} has the following property: for any motion of the fluid there is a function $p(t,\vf{x})$ called the \emph{pressure} such that if $S$ is a surface in the fluid with a chosen unit normal $\vf{n}$, the force of stress exerted across the surface $S$ per unit area at $\vf{x}\in S$ at time $t$ is $p(t,\vf{x})\vf{n}$. Thus, the total force of stress exerted inside a region $W\subseteq D$ is given by: + An \emph{ideal fluid} has the following property: for any motion of the fluid there is a function $p(\vf{x},t)$ called the \emph{pressure} such that if $S$ is a surface in the fluid with a chosen unit normal $\vf{n}$, the force of stress exerted across the surface $S$ per unit area at $\vf{x}\in S$ at time $t$ is $p(\vf{x},t)\vf{n}$. Thus, the total force of stress exerted inside a region $W\subseteq D$ is given by: $$ \vf{A}_{\partial W}:=\text{Force on $W$}=-\int_{\Fr{W}}p\vf{n}\dd{S} $$ @@ -86,5 +86,243 @@ \end{align*} Integrating over $W$ and using the \mnameref{FSV:divergencethm;PDE:fundamentallema} we obtain the result. \end{proof} + \begin{definition} + Let $\vf{x}\in D$. We denote by $\vf\varphi(\vf{x},t)$ the position of the fluid particle $\vf{x}$ at time $t$ and fixed $t\in \RR$, $\vf\varphi_t:\vf{x}\to\vf\varphi(\vf{x},t)$. If $W\subseteq D$, we denote $W_t:=\vf\varphi_t(W)$ the volume $W$ moving with the fluid. + \end{definition} + \begin{lemma}\label{FLM:lemmaJacobian} + Let $J(\vf{x},t)$ be the Jacobian determinant of $\vf\varphi_t$. Then: + $$ + \pdv{}{t} J(\vf{x},t)=J(\vf{x},t)(\div\vf{u})(\vf\varphi(\vf{x},t),t) + $$ + \end{lemma} + \begin{proof} + We have that $J=\det\vf{D\varphi} = \det(\pdv{\phi_1}{\vf{x}},\pdv{\phi_2}{\vf{x}},\pdv{\phi_3}{\vf{x}})$, where $\vf\varphi = (\phi_1,\phi_2,\phi_3)$ and $\pdv{\phi_i}{\vf{x}} := \transpose{\left( + \pdv{\phi_i}{x}, \pdv{\phi_i}{y}, \pdv{\phi_i}{z} + \right)}$. Hence, from the multilineary property of the determinant we have: + \begin{multline}\label{FLM:Jacobian} + \pdv{}{t}J=\det(\pdv{}{t}\pdv{\phi_1}{\vf{x}},\pdv{\phi_2}{\vf{x}},\pdv{\phi_3}{\vf{x}})+\det(\pdv{\phi_1}{\vf{x}},\pdv{}{t}\pdv{\phi_2}{\vf{x}},\pdv{\phi_3}{\vf{x}})\\+\det(\pdv{\phi_1}{\vf{x}},\pdv{\phi_2}{\vf{x}},\pdv{}{t}\pdv{\phi_3}{\vf{x}}) + \end{multline} + Now if $\vf{u}=(u_1,u_2,u_3)$, then: + \begin{align*} + \pdv{}{t}\pdv{\phi_i}{\vf{x}} & =\pdv{}{\vf{x}}u_i(\vf\varphi(\vf{x},t),t) \\ + & =\pdv{u_i}{\phi_1}\pdv{\phi_1}{\vf{x}}+\pdv{u_i}{\phi_2}\pdv{\phi_2}{\vf{x}}+\pdv{u_i}{\phi_3}\pdv{\phi_3}{\vf{x}} \\ + \end{align*} + because $\pdv{\phi_i}{t} = u_i(\vf\varphi(\vf{x},t),t)$. + Introducing this into \mcref{FLM:Jacobian} we obtain: + $$ + \pdv{}{t}J=J\left(\pdv{u_1}{\phi_1}+\pdv{u_2}{\phi_2}+\pdv{u_3}{\phi_3}\right)=J(\div\vf{u})(\vf\varphi(\vf{x},t),t) + $$ + \end{proof} + \begin{corollary} + We have: + $$ + \dv{}{t}\int_{W_t}\rho\vf{u}\dd{V}=\int_{W_t}\rho\matdv{\vf{u}}{t}\dd{V} + $$ + \end{corollary} + \begin{proof} + Using the \mnameref{FSV:changeofvariable,FLM:lemmaJacobian} we have that: + \begin{align*} + \begin{split} + \dv{}{t}\int_{W_t}\rho\vf{u}\dd{V}&=\int_W\left[ + \matdv{}{t}(\rho\vf{u})(\vf\varphi(\vf{x},t),t)+(\rho\vf{u})\cdot\right.\\ + &\hspace{2cm}\cdot(\div\vf{u})(\vf\varphi(\vf{x},t),t) + \bigg]J(\vf{x},t)\dd{V} + \end{split} \\ + & = \int_{W_t}\matdv{}{t}(\rho\vf{u})+(\rho\div\vf{u}) \vf{u}\dd{V} \\ + & = \int_{W_t}\rho\matdv{\vf{u}}{t}\dd{V} + \end{align*} + where the last equality follows from the \mcref{FLM:conservationofmass}: + $$ + \matdv{\rho}{t}+\rho\div\vf{u}=\pdv{\rho}{t}+\div(\rho\vf{u})=0 + $$ + \end{proof} + \begin{corollary}[Transport theorem]\label{FLM:trasport} + For any smooth enough function $f(\vf{x},t)$ we have: + \begin{align*} + \dv{}{t}\int_{W_t}\rho f\dd{V} & =\int_{W_t}\rho\matdv{f}{t}\dd{V} \\ + \dv{}{t}\int_{W_t} f\dd{V} & =\int_{W_t}\left[\dv{f}{t}+\div(f\vf{u})\right]\dd{V} \\ + \end{align*} + \end{corollary} + \begin{definition} + A flow is called \emph{incompressible} if for any fluid subregion $W\subseteq D$ we have: + $$ + \vol(W_t)=\vol(W)=\const + $$ + Otherwise, the flow is called \emph{compressible}. + \end{definition} + \begin{proposition}\label{FLM:incompressible_eq} + Consider the flow $\vf\varphi$ and its Jacobian $J$. Then, the following are equivalent: + \begin{enumerate} + \item The flow is incompressible. + \item $\div\vf{u}=0$. + \item $J=1$. + \end{enumerate} + \end{proposition} + \begin{proof} + Note that: + $$ + \dv{}{t}\int_{W_t}\dd{V}=\dv{}{t}\int_WJ\dd{V}=\int_{W_t}\div\vf{u}\dd{V} + $$ + Hence, if $\div \vf{u}=0$ then the flow is incompressible. Now, if the flow is incompressible we have that: + $$ + 0=\dv{}{t}\int_{W_t}\dd{V}=\int_W\dv{J}{t}\dd{V} + $$ + which is implies that $J=\const$ by \mnameref{PDE:fundamentallema}. Since $J(\vf{x},0)=1$ we have that $J=1$. Finally, from \mcref{FLM:lemmaJacobian} we have that if $J=1$ then $\div\vf{u}=0$. + \end{proof} + \begin{definition} + A fluid is called \emph{homogeneous} if $\rho=\rho(t)$, that is, if $\rho$ is constant in space. + \end{definition} + \begin{proposition} + A fluid is incompressible if and only if $\matdv{\rho}{t}=0$. In particular, if the fluid is homogeneous, then it is incompressible if and only if $\rho=\const$ (i.e.\ it is also constant in time). + \end{proposition} + \begin{proof} + We can write \mcref{FLM:continuityequation} as: + $$ + \matdv{\rho}{t}+\rho\div\vf{u}=0 + $$ + And the result follows from \mcref{FLM:incompressible_eq}. + \end{proof} + \begin{proposition} + Let $J$ be the Jacobian of the flow $\vf\varphi$. Then: + $$ + \rho(\vf\varphi(\vf{x},t),t)J(\vf{x},t)=\rho(\vf{x},0) + $$ + \end{proposition} + \begin{sproof} + From \mnameref{FLM:trasport} with $f=1$ we have: + $$ + \int_{W_0}\rho(\vf{x},0)\dd{V}=\int_{W_t}\rho\dd{V}=\dv{}{t}\int_{W_0}\rho J\dd{V} + $$ + Since, $W_0$ is arbitrary, the result follows from \mnameref{PDE:fundamentallema}. + \end{sproof} + \begin{remark} + As a corollary, a fluid that is homogeneous at $t = 0$ but is compressible, will generally not remain homogeneous. However, the fluid will remain homogeneous if it is incompressible. + \end{remark} + \begin{definition} + The \emph{kinetic energy} of a moving portion $W_t$ of a fluid is defined as: + $$ + E_\mathrm{kinetic}= \frac{1}{2}\int_{W_t}\rho\norm{\vf{u}}^2\dd{V} + $$ + where the norm is the Euclidean norm. + \end{definition} + \begin{lemma}\label{FLM:lemmaEkinetic} + The rate of change of kinetic energy is given by: + $$ + \dv{E_\mathrm{kinetic}}{t}=\int_{W_t}\rho\vf{u}\cdot\matdv{\vf{u}}{t}\dd{V} + $$ + \end{lemma} + \begin{proof} + From \mnameref{FLM:trasport} we have that: + $$ + \dv{E_\mathrm{kinetic}}{t}= \frac{1}{2}\int_{W_t}\rho\matdv{\norm{\vf{u}}^2}{t}\dd{V} + $$ + Now use the linearity of the material derivative and the dot product. + \end{proof} + \begin{theorem} + Consider an incompressible fluid such that the rate of change of kinetic energy in a portion of fluid equals the rate at which the pressure and body forces do work: + $$ + \dv{E_\mathrm{kinetic}}{t}=-\int_{\partial W_t}p\vf{u}\cdot\dd{\vf{S}}+\int_{W_t}\rho\vf{u}\cdot\vf{f}\dd{V} + $$ + Then, the Euler equations that completely describe the motion of the fluid are: + $$ + \begin{cases} + \displaystyle\rho \matdv{\vf{u}}{t}= -\grad p + \rho\vf{f} \\ + \displaystyle\matdv{\rho}{t}=0 \\ + \displaystyle\div\vf{u}=0 + \end{cases} + $$ + with the boundary conditions $\vf{u}\cdot\vf{n} = 0$ on $\partial D$. + \end{theorem} + \begin{proof} + From \mnameref{FLM:lemmaEkinetic} and using the \mnameref{FSV:divergencethm} we have: + \begin{align*} + \int_{W_t}\rho\vf{u}\cdot\matdv{\vf{u}}{t}\dd{V} & =-\int_{W_t}\left[\div(p\vf{u})-\rho \vf{u}\cdot\vf{f}\right]\dd{V} \\ + & = -\int_{W_t}\left[\vf{u}\cdot\grad p - \rho \vf{u}\cdot\vf{f}\right] + \end{align*} + because $\div\vf{u}=0$. This equation is a consequence of balance of momentum. + \end{proof} + \begin{remark} + This argument, in addition, shows that if we assume $E = E_\mathrm{kinetic}$, then the fluid must be incompressible. + \end{remark} + \begin{definition} + A compressible flow is called \emph{isentropic} if there exists a function $w$, called the \emph{enthalpy}, such that: + $$ + \grad w=\frac{1}{\rho}\grad p + $$ + \end{definition} + \begin{remark} + From this part we will need some basic concepts of thermodynamics, that we review here. Recall that: + \begin{gather*} + p=\text{pressure} \ \ \rho=\text{density} \ \ T=\text{temperature} \ \ s=\text{entropy} \\ + w=\text{enthalpy} \ \ \epsilon=\text{internal energy per unit mass} + \end{gather*} + These quantities are related by the First Law of Thermodynamics: + \begin{equation}\label{FLM:firstlawthermo} + \dd{w}=T\dd{s}+\frac{1}{\rho}\dd{p} + \end{equation} + which using that $\epsilon=w-p/\rho$ can be written as: + $$ + \dd{\epsilon}=T\dd{s}+\frac{p}{\rho^2}\dd{\rho} + $$ + \end{remark} + \begin{remark} + Note that if the pressure is a function of $\rho$ only, then the flow is isentropic by defining $w=\int \frac{p'(\rho)}{\rho} \dd{\rho}$ which is the integrated version of \mcref{FLM:firstlawthermo}. + \end{remark} + \begin{theorem} + For isentropic flows, the integral form of the energy balace reads as follows: The rate of change of energy in a portion of fluid equals the rate at which work is done on it. + \begin{align*} + \dv{E_\mathrm{total}}{t} & =\dv{}{t}\int_{W_t}\left[\frac{1}{2} \rho\norm{\vf{u}}^2+\rho \epsilon\right]\dd{V} \\ + & =\int_{W_t}\rho\vf{u}\cdot \vf{f} \dd{V}-\int_{\partial W_t}p\vf{u}\cdot\dd{\vf{S}} + \end{align*} + And the Euler equations are: + $$ + \begin{cases} + \displaystyle\matdv{\vf{u}}{t}= -\grad w + \vf{f} \\ + \displaystyle\dv{\rho}{t}+\div(\rho\vf{u})=0 + \end{cases} + $$ + and the boundary conditions are $\vf{u}\cdot\vf{n}=0$ on $\partial D$. + \end{theorem} + \begin{remark} + Gases can often be treated as isentropic fluid with $p=A\rho^\gamma$ where $A$ and $\gamma\geq 1$ are constants. Here: + $$ + w = \frac{\gamma A \rho^{\gamma-1}}{\gamma-1}\quad \epsilon = \frac{A\rho^{\gamma-1}}{\gamma-1} + $$ + \end{remark} + \begin{definition} + Given a fluid with velocity field $\vf{u}(\vf{x}, t)$, a \emph{streamline} is a curve $\vf{x}(s)$ such that $\vf{u}(\vf{x}(s), t)=\dv{\vf{x}}{s}$ with $t$ fixed. + \end{definition} + \begin{definition} + We define the trajectory as the curve $\vf{x}(t)$ such that $\vf{u}(\vf{x}(t), t)=\dv{\vf{x}}{t}$. + \end{definition} + \begin{remark} + If $\vf{u}$ is independent of $t$, then the streamlines and trajectories coincide. In this case, the fluid is said to be \emph{stationary} or \emph{steady}. + \end{remark} + \begin{theorem}[Bernoulli's theorem] + In a stationary isentropic flow with a present conservative force $\vf{f}=-\grad\psi$, the quantity + $$ + \frac{1}{2}\norm{\vf{u}}^2+w+\psi + $$ + is constant along streamlines. The same holds for homogeneous incompressible flow with $w$ replaced by $p/\rho$. + \end{theorem} + \begin{proof} + An easy check shows that: + $$ + \frac{1}{2}\grad(\norm{\vf{u}}^2)=(\vf{u}\cdot\grad)\vf{u}+\vf{u}\times(\rotp\vf{u}) + $$ + Because the flow is steady, the equations of motion give $(\vf{u}\cdot\grad)\vf{u}=-\grad w+\vf{f}$. Thus: + $$ + \grad\left(\frac{1}{2}\norm{\vf{u}}^2+w+\psi\right)=\vf{u}\times (\rotp\vf{u}) + $$ + Let $\vf{x}(s)$ be a streamline. Then: + \begin{equation*} + \dv{}{s}\left[\left(\frac{1}{2}\norm{\vf{u}}^2+w+\psi\right)(\vf{x}(s),t)\right]\! =\! [\vf{u}\times (\rotp\vf{u})]\cdot\vf{x}'(s)=0 + \end{equation*} + because $\vf{x}'(s)=\vf{u}$ is orthogonal to $\vf{u}\times (\rotp\vf{u})$. + \end{proof} + \subsubsection{Rotation and vorticity} + \begin{definition} + Let $\vf{u}=(u,v,w)$ be the velocity field of a fluid. The \emph{vorticity} is the vector field $\vf{\omega}:=\rotp\vf{u}$. + \end{definition} \end{multicols} \end{document} \ No newline at end of file diff --git a/index.html b/index.html index a6887ce..87156d5 100644 --- a/index.html +++ b/index.html @@ -66,37 +66,35 @@

Mathematics

-

Physics

-

Full summaries of

+

Topics on Physics

+ +
diff --git a/main_math.idx b/main_math.idx index 2f629b9..e55f700 100644 --- a/main_math.idx +++ b/main_math.idx @@ -2649,6 +2649,8 @@ \indexentry{FTBS|hyperpage}{267} \indexentry{Forward-time central-space|hyperpage}{267} \indexentry{FTCS|hyperpage}{267} +\indexentry{Backward-time central-space|hyperpage}{267} +\indexentry{BTCS|hyperpage}{267} \indexentry{Leapfrog scheme|hyperpage}{267} \indexentry{Lax-Friedrichs scheme|hyperpage}{267} \indexentry{stability region|hyperpage}{267} @@ -2658,7 +2660,7 @@ \indexentry{conditional consistency|hyperpage}{268} \indexentry{upwind condition|hyperpage}{268} \indexentry{stable|hyperpage}{268} -\indexentry{Courant-Friedrichs-Lewy condition|hyperpage}{268} +\indexentry{Courant-Friedrichs-Lewy condition|hyperpage}{269} \indexentry{Semidiscrete Fourier transform|hyperpage}{269} \indexentry{semidiscrete Fourier transform|hyperpage}{269} \indexentry{inverse semidiscrete Fourier transform|hyperpage}{269} @@ -2666,380 +2668,388 @@ \indexentry{amplification factor|hyperpage}{269} \indexentry{Lax-Wendroff|hyperpage}{270} \indexentry{Lax-Wendroff scheme|hyperpage}{270} -\indexentry{Crank-Nicolson|hyperpage}{270} -\indexentry{Crank-Nicolson scheme|hyperpage}{270} +\indexentry{Crank-Nicolson|hyperpage}{271} +\indexentry{Crank-Nicolson scheme|hyperpage}{271} \indexentry{numerical boundary condition|hyperpage}{271} -\indexentry{Lax-Richtmyer equivalence theorem|hyperpage}{271} +\indexentry{Lax-Richtmyer equivalence theorem|hyperpage}{272} \indexentry{amplification polynomial|hyperpage}{272} \indexentry{compatibility condition|hyperpage}{272} \indexentry{elliptic|hyperpage}{272} \indexentry{hyperbolic|hyperpage}{272} \indexentry{parabolic|hyperpage}{272} -\indexentry{stable|hyperpage}{272} -\indexentry{Forward-time central-space|hyperpage}{272} -\indexentry{Backward-time central-space|hyperpage}{272} -\indexentry{Crank-Nicolson scheme|hyperpage}{272} -\indexentry{Leapfrog scheme|hyperpage}{272} -\indexentry{Du-Fort-Frankel scheme|hyperpage}{272} +\indexentry{stable|hyperpage}{273} +\indexentry{Forward-time central-space|hyperpage}{273} +\indexentry{Backward-time central-space|hyperpage}{273} +\indexentry{Crank-Nicolson scheme|hyperpage}{273} +\indexentry{Leapfrog scheme|hyperpage}{273} +\indexentry{Du-Fort-Frankel scheme|hyperpage}{273} \indexentry{Dirichlet|hyperpage}{273} \indexentry{Neumann|hyperpage}{273} \indexentry{Robin|hyperpage}{273} \indexentry{discrete laplacian|hyperpage}{273} \indexentry{Discrete maximum principle|hyperpage}{273} \indexentry{Discrete minimum principle|hyperpage}{273} -\indexentry{finite element method|hyperpage}{273} +\indexentry{finite element method|hyperpage}{274} \indexentry{variational formulation|hyperpage}{274} +\indexentry{weak formulation|hyperpage}{274} \indexentry{Galerkin approximation|hyperpage}{274} \indexentry{Galerkin approximation|hyperpage}{274} \indexentry{Mesh|hyperpage}{274} \indexentry{mesh|hyperpage}{274} \indexentry{cells|hyperpage}{274} \indexentry{finite element method|hyperpage}{274} +\indexentry{FEM|hyperpage}{274} \indexentry{nodes|hyperpage}{274} \indexentry{nodes|hyperpage}{274} \indexentry{nodal basis|hyperpage}{274} -\indexentry{Wave equation|hyperpage}{275} -\indexentry{hyperbolic equations|hyperpage}{275} -\indexentry{Navier-Cauchy equation|hyperpage}{275} -\indexentry{Lamé coefficients|hyperpage}{275} -\indexentry{elastodynamics|hyperpage}{275} -\indexentry{material derivative operator|hyperpage}{275} -\indexentry{incompressible flow|hyperpage}{275} -\indexentry{Continuous equation|hyperpage}{275} -\indexentry{continuous equation|hyperpage}{275} -\indexentry{Cauchy momentum equation|hyperpage}{275} -\indexentry{Cauchy momentum equation|hyperpage}{275} -\indexentry{Inviscid flow|hyperpage}{275} -\indexentry{Viscid flow|hyperpage}{275} -\indexentry{potential|hyperpage}{275} -\indexentry{Poisson equation|hyperpage}{275} -\indexentry{elliptic equation|hyperpage}{275} -\indexentry{Fick's law of diffusion|hyperpage}{276} -\indexentry{diffusivity|hyperpage}{276} -\indexentry{diffusion coefficient|hyperpage}{276} -\indexentry{diffusion flux|hyperpage}{276} -\indexentry{Fick's law|hyperpage}{276} -\indexentry{Diffusion equation|hyperpage}{276} -\indexentry{Fourier's law|hyperpage}{276} -\indexentry{thermal conductivity|hyperpage}{276} -\indexentry{heat flux|hyperpage}{276} -\indexentry{Fourier's law|hyperpage}{276} -\indexentry{Heat equation|hyperpage}{276} -\indexentry{specific heat capacity|hyperpage}{276} -\indexentry{density|hyperpage}{276} -\indexentry{thermal diffusivity|hyperpage}{276} -\indexentry{Gau\ss ' law|hyperpage}{276} -\indexentry{Gau\ss ' law|hyperpage}{276} -\indexentry{Gau\ss ' law for magnetism|hyperpage}{276} -\indexentry{Gau\ss ' law for magnetism|hyperpage}{276} -\indexentry{Maxwell-Faraday equation|hyperpage}{276} -\indexentry{Maxwell-Faraday equation|hyperpage}{276} -\indexentry{Ampère-Maxwell circuital law|hyperpage}{276} -\indexentry{Ampère's law|hyperpage}{276} -\indexentry{displacement current|hyperpage}{276} -\indexentry{refractive index|hyperpage}{276} -\indexentry{Fermat's principle|hyperpage}{276} -\indexentry{Fermat's principle|hyperpage}{276} -\indexentry{Eikonal equation|hyperpage}{276} -\indexentry{action|hyperpage}{276} -\indexentry{Principle of least action|hyperpage}{277} -\indexentry{small change|hyperpage}{277} -\indexentry{Hamilton-Jacobi equation|hyperpage}{277} -\indexentry{Schrödinger equation|hyperpage}{277} -\indexentry{Schrödinger equation|hyperpage}{277} -\indexentry{wave function|hyperpage}{277} -\indexentry{Fundamental lemma of calculus of variations|hyperpage}{277} -\indexentry{Green identities|hyperpage}{277} -\indexentry{Method of characteristics|hyperpage}{277} -\indexentry{characteristic curves|hyperpage}{277} -\indexentry{Traffic flow equation|hyperpage}{278} -\indexentry{Rankine-Hugoniot equation|hyperpage}{278} -\indexentry{Entropy condition|hyperpage}{278} -\indexentry{D'Alembert formula|hyperpage}{279} -\indexentry{Conservation of energy|hyperpage}{280} -\indexentry{even periodic extension|hyperpage}{281} -\indexentry{odd periodic extension|hyperpage}{281} -\indexentry{Separation of variables|hyperpage}{281} -\indexentry{separation of variables|hyperpage}{281} -\indexentry{Sturm-Picone comparison theorem|hyperpage}{281} -\indexentry{self-similar|hyperpage}{282} -\indexentry{Distribution|hyperpage}{283} -\indexentry{distribution|hyperpage}{283} -\indexentry{Dirac's $\delta $ distribution|hyperpage}{283} -\indexentry{differentiation operator|hyperpage}{283} -\indexentry{distributional derivative|hyperpage}{283} -\indexentry{Heaviside step function|hyperpage}{283} -\indexentry{fundamental solution|hyperpage}{283} -\indexentry{heat kernel|hyperpage}{283} -\indexentry{generalized heat kernel|hyperpage}{284} -\indexentry{Explicit scheme in finite differences|hyperpage}{284} -\indexentry{Implicit scheme in finite differences|hyperpage}{284} -\indexentry{Duhamel principle|hyperpage}{285} -\indexentry{parabolic cylinder|hyperpage}{285} -\indexentry{parabolic boundary|hyperpage}{285} -\indexentry{Maximum principle|hyperpage}{285} -\indexentry{Minimum principle|hyperpage}{285} -\indexentry{Uniqueness of the heat equation|hyperpage}{285} -\indexentry{Maximum principle on unbounded domains|hyperpage}{285} -\indexentry{Minimum principle on unbounded domains|hyperpage}{286} -\indexentry{Uniqueness of the heat equation on the unbounded domains|hyperpage}{286} -\indexentry{Laplace equation|hyperpage}{286} -\indexentry{Laplace equation|hyperpage}{286} -\indexentry{Dirichlet problem in the disc|hyperpage}{286} -\indexentry{Dirichlet problem|hyperpage}{286} -\indexentry{Dirichlet problem|hyperpage}{286} -\indexentry{Uniqueness of Dirichlet problem|hyperpage}{286} -\indexentry{energy functional|hyperpage}{286} -\indexentry{Dirichlet's principle|hyperpage}{286} -\indexentry{Sobolev space|hyperpage}{287} -\indexentry{average|hyperpage}{287} -\indexentry{Trace theorem|hyperpage}{287} -\indexentry{trace|hyperpage}{287} -\indexentry{Poincaré inequality|hyperpage}{287} -\indexentry{$\sigma $-algebra|hyperpage}{289} -\indexentry{$\sigma $-algebra|hyperpage}{289} -\indexentry{Measure|hyperpage}{289} -\indexentry{measure|hyperpage}{289} -\indexentry{$\sigma $-additivity|hyperpage}{289} -\indexentry{interval|hyperpage}{289} -\indexentry{volume|hyperpage}{289} -\indexentry{$m$-th dyadic cube|hyperpage}{289} -\indexentry{Outer measure|hyperpage}{290} -\indexentry{outer measure|hyperpage}{290} -\indexentry{null set|hyperpage}{291} -\indexentry{almost everywhere|hyperpage}{291} -\indexentry{a.e.|hyperpage}{291} -\indexentry{Lebesgue measure|hyperpage}{291} -\indexentry{Lebesgue measurable|hyperpage}{291} -\indexentry{measurable|hyperpage}{291} -\indexentry{Lebesgue measure|hyperpage}{291} -\indexentry{real function|hyperpage}{292} -\indexentry{finite|hyperpage}{292} +\indexentry{stiffness matrix|hyperpage}{275} +\indexentry{load vector|hyperpage}{275} +\indexentry{Wave equation|hyperpage}{276} +\indexentry{hyperbolic equations|hyperpage}{276} +\indexentry{Navier-Cauchy equation|hyperpage}{276} +\indexentry{Lamé coefficients|hyperpage}{276} +\indexentry{elastodynamics|hyperpage}{276} +\indexentry{material derivative operator|hyperpage}{276} +\indexentry{incompressible flow|hyperpage}{276} +\indexentry{Continuous equation|hyperpage}{276} +\indexentry{continuous equation|hyperpage}{276} +\indexentry{Cauchy momentum equation|hyperpage}{276} +\indexentry{Cauchy momentum equation|hyperpage}{276} +\indexentry{Inviscid flow|hyperpage}{276} +\indexentry{Viscid flow|hyperpage}{276} +\indexentry{potential|hyperpage}{276} +\indexentry{Poisson equation|hyperpage}{276} +\indexentry{elliptic equation|hyperpage}{276} +\indexentry{Fick's law of diffusion|hyperpage}{277} +\indexentry{diffusivity|hyperpage}{277} +\indexentry{diffusion coefficient|hyperpage}{277} +\indexentry{diffusion flux|hyperpage}{277} +\indexentry{Fick's law|hyperpage}{277} +\indexentry{Diffusion equation|hyperpage}{277} +\indexentry{Fourier's law|hyperpage}{277} +\indexentry{thermal conductivity|hyperpage}{277} +\indexentry{heat flux|hyperpage}{277} +\indexentry{Fourier's law|hyperpage}{277} +\indexentry{Heat equation|hyperpage}{277} +\indexentry{specific heat capacity|hyperpage}{277} +\indexentry{density|hyperpage}{277} +\indexentry{thermal diffusivity|hyperpage}{277} +\indexentry{Gau\ss ' law|hyperpage}{277} +\indexentry{Gau\ss ' law|hyperpage}{277} +\indexentry{Gau\ss ' law for magnetism|hyperpage}{277} +\indexentry{Gau\ss ' law for magnetism|hyperpage}{277} +\indexentry{Maxwell-Faraday equation|hyperpage}{277} +\indexentry{Maxwell-Faraday equation|hyperpage}{277} +\indexentry{Ampère-Maxwell circuital law|hyperpage}{277} +\indexentry{Ampère's law|hyperpage}{277} +\indexentry{displacement current|hyperpage}{277} +\indexentry{refractive index|hyperpage}{277} +\indexentry{Fermat's principle|hyperpage}{277} +\indexentry{Fermat's principle|hyperpage}{277} +\indexentry{Eikonal equation|hyperpage}{277} +\indexentry{action|hyperpage}{277} +\indexentry{Principle of least action|hyperpage}{278} +\indexentry{small change|hyperpage}{278} +\indexentry{Hamilton-Jacobi equation|hyperpage}{278} +\indexentry{Schrödinger equation|hyperpage}{278} +\indexentry{Schrödinger equation|hyperpage}{278} +\indexentry{wave function|hyperpage}{278} +\indexentry{Fundamental lemma of calculus of variations|hyperpage}{278} +\indexentry{Green identities|hyperpage}{278} +\indexentry{Method of characteristics|hyperpage}{278} +\indexentry{characteristic curves|hyperpage}{278} +\indexentry{Traffic flow equation|hyperpage}{279} +\indexentry{Rankine-Hugoniot equation|hyperpage}{279} +\indexentry{Entropy condition|hyperpage}{279} +\indexentry{D'Alembert formula|hyperpage}{280} +\indexentry{Conservation of energy|hyperpage}{281} +\indexentry{even periodic extension|hyperpage}{282} +\indexentry{odd periodic extension|hyperpage}{282} +\indexentry{Separation of variables|hyperpage}{282} +\indexentry{separation of variables|hyperpage}{282} +\indexentry{Sturm-Picone comparison theorem|hyperpage}{282} +\indexentry{self-similar|hyperpage}{283} +\indexentry{Distribution|hyperpage}{284} +\indexentry{distribution|hyperpage}{284} +\indexentry{Dirac's $\delta $ distribution|hyperpage}{284} +\indexentry{differentiation operator|hyperpage}{284} +\indexentry{distributional derivative|hyperpage}{284} +\indexentry{Heaviside step function|hyperpage}{284} +\indexentry{fundamental solution|hyperpage}{284} +\indexentry{heat kernel|hyperpage}{284} +\indexentry{generalized heat kernel|hyperpage}{285} +\indexentry{Explicit scheme in finite differences|hyperpage}{285} +\indexentry{Implicit scheme in finite differences|hyperpage}{285} +\indexentry{Duhamel principle|hyperpage}{286} +\indexentry{parabolic cylinder|hyperpage}{286} +\indexentry{parabolic boundary|hyperpage}{286} +\indexentry{Maximum principle|hyperpage}{286} +\indexentry{Minimum principle|hyperpage}{286} +\indexentry{Uniqueness of the heat equation|hyperpage}{286} +\indexentry{Maximum principle on unbounded domains|hyperpage}{286} +\indexentry{Minimum principle on unbounded domains|hyperpage}{287} +\indexentry{Uniqueness of the heat equation on the unbounded domains|hyperpage}{287} +\indexentry{Laplace equation|hyperpage}{287} +\indexentry{Laplace equation|hyperpage}{287} +\indexentry{Dirichlet problem in the disc|hyperpage}{287} +\indexentry{Dirichlet problem|hyperpage}{287} +\indexentry{Dirichlet problem|hyperpage}{287} +\indexentry{Uniqueness of Dirichlet problem|hyperpage}{287} +\indexentry{energy functional|hyperpage}{287} +\indexentry{Dirichlet's principle|hyperpage}{287} +\indexentry{Sobolev space|hyperpage}{288} +\indexentry{average|hyperpage}{288} +\indexentry{Trace theorem|hyperpage}{288} +\indexentry{trace|hyperpage}{288} +\indexentry{Poincaré inequality|hyperpage}{288} +\indexentry{$\sigma $-algebra|hyperpage}{290} +\indexentry{$\sigma $-algebra|hyperpage}{290} +\indexentry{Measure|hyperpage}{290} +\indexentry{measure|hyperpage}{290} +\indexentry{$\sigma $-additivity|hyperpage}{290} +\indexentry{interval|hyperpage}{290} +\indexentry{volume|hyperpage}{290} +\indexentry{$m$-th dyadic cube|hyperpage}{290} +\indexentry{Outer measure|hyperpage}{291} +\indexentry{outer measure|hyperpage}{291} +\indexentry{null set|hyperpage}{292} +\indexentry{almost everywhere|hyperpage}{292} +\indexentry{a.e.|hyperpage}{292} +\indexentry{Lebesgue measure|hyperpage}{292} \indexentry{Lebesgue measurable|hyperpage}{292} \indexentry{measurable|hyperpage}{292} -\indexentry{Borel $\sigma $-algebra|hyperpage}{293} -\indexentry{Borel measurable|hyperpage}{293} -\indexentry{simple function|hyperpage}{293} -\indexentry{integral of $s$ over $\ensuremath {\mathbb {R}}^n$|hyperpage}{293} -\indexentry{integral of $s$ over a measurable set $E$|hyperpage}{293} -\indexentry{integral of $f$ over $\ensuremath {\mathbb {R}}^n$|hyperpage}{294} -\indexentry{integral of $f$ over a measurable set $E\subseteq \ensuremath {\mathbb {R}}^n$|hyperpage}{294} -\indexentry{Monotone convergence theorem|hyperpage}{294} -\indexentry{Chebyshev's inequality|hyperpage}{295} -\indexentry{Fatou's lemma|hyperpage}{295} -\indexentry{integral of $f$ over $E$|hyperpage}{295} -\indexentry{integrable function over $E$|hyperpage}{295} -\indexentry{Dominated convergence theorem|hyperpage}{295} -\indexentry{converge in mean|hyperpage}{296} -\indexentry{exists and it is finite|hyperpage}{296} -\indexentry{Mean value theorem for integrals|hyperpage}{296} -\indexentry{Barrow's law|hyperpage}{296} -\indexentry{Fundamental theorem of calculus|hyperpage}{296} -\indexentry{Integration by parts|hyperpage}{296} -\indexentry{Hardy-Littlewood maximal function|hyperpage}{296} -\indexentry{Hardy-Littlewood maximal function|hyperpage}{296} -\indexentry{Lebesgue differentiation theorem|hyperpage}{297} -\indexentry{section|hyperpage}{297} -\indexentry{Tonelli's theorem|hyperpage}{297} -\indexentry{Fubini's theorem|hyperpage}{298} -\indexentry{change of variables|hyperpage}{298} -\indexentry{Change of variables|hyperpage}{298} -\indexentry{distance|hyperpage}{298} -\indexentry{triangular inequality|hyperpage}{298} -\indexentry{metric space|hyperpage}{298} -\indexentry{complete|hyperpage}{298} -\indexentry{norm|hyperpage}{298} -\indexentry{triangular inequality|hyperpage}{298} -\indexentry{normed vector space|hyperpage}{298} -\indexentry{convergent series|hyperpage}{298} -\indexentry{absolutely convergent|hyperpage}{298} -\indexentry{Banach space|hyperpage}{299} -\indexentry{Banach space|hyperpage}{299} -\indexentry{uniform norm|hyperpage}{299} -\indexentry{total subset|hyperpage}{299} -\indexentry{separable|hyperpage}{299} -\indexentry{quotient space|hyperpage}{300} -\indexentry{Young's inequality for products|hyperpage}{300} -\indexentry{Hölder conjugates|hyperpage}{300} -\indexentry{Hölder's inequality|hyperpage}{300} -\indexentry{Minkowski inequality|hyperpage}{301} -\indexentry{uniform norm|hyperpage}{301} -\indexentry{uniform norm|hyperpage}{301} -\indexentry{supremum norm|hyperpage}{301} -\indexentry{subalgebra|hyperpage}{301} -\indexentry{separating set|hyperpage}{301} -\indexentry{separate the points|hyperpage}{301} -\indexentry{vanishes nowhere|hyperpage}{301} -\indexentry{self-conjugate|hyperpage}{302} -\indexentry{Stone-Weierstra\ss \ theorem|hyperpage}{302} -\indexentry{pointwise bounded|hyperpage}{302} -\indexentry{locally bounded|hyperpage}{302} -\indexentry{uniformly bounded|hyperpage}{302} -\indexentry{equicontinuous at a point|hyperpage}{302} -\indexentry{pointwise equicontinuous|hyperpage}{302} -\indexentry{uniformly equicontinuous|hyperpage}{302} -\indexentry{Arzelà-Ascoli theorem|hyperpage}{302} -\indexentry{operator|hyperpage}{303} -\indexentry{norm|hyperpage}{303} -\indexentry{sublinear|hyperpage}{304} -\indexentry{Marcinkiewicz interpolation theorem|hyperpage}{304} -\indexentry{topological homeomorphism|hyperpage}{304} -\indexentry{isomorphic|hyperpage}{304} -\indexentry{is finer than|hyperpage}{304} -\indexentry{equivalent|hyperpage}{304} -\indexentry{Almost orthogonality lemma|hyperpage}{304} -\indexentry{Riesz's theorem|hyperpage}{304} -\indexentry{normed algebra|hyperpage}{305} -\indexentry{dual space|hyperpage}{305} -\indexentry{compact operator|hyperpage}{305} -\indexentry{Fredholm operator with kernel $K$|hyperpage}{305} -\indexentry{Volterra operator with kernel $K$|hyperpage}{305} -\indexentry{Hilbert-Schmidt operator with kernel $K$|hyperpage}{305} -\indexentry{finite-rank operator|hyperpage}{305} -\indexentry{Neumann series|hyperpage}{306} -\indexentry{convex functional|hyperpage}{306} -\indexentry{Hahn-Banach theorem|hyperpage}{306} -\indexentry{prolongation|hyperpage}{306} -\indexentry{Seminorm|hyperpage}{306} -\indexentry{seminorm|hyperpage}{306} -\indexentry{Hahn-Banach theorem|hyperpage}{306} -\indexentry{Hahn-Banach theorem|hyperpage}{306} -\indexentry{reflexive|hyperpage}{306} -\indexentry{dual map|hyperpage}{306} -\indexentry{proper subspace of $T$|hyperpage}{307} -\indexentry{eigenvectors|hyperpage}{307} -\indexentry{eigenvalue|hyperpage}{307} -\indexentry{spectrum|hyperpage}{307} -\indexentry{spectral values|hyperpage}{307} -\indexentry{Baire's theorem|hyperpage}{307} -\indexentry{Open mapping theorem|hyperpage}{307} -\indexentry{Closed graph theorem|hyperpage}{307} -\indexentry{Banach-Steinhaus theorem|hyperpage}{307} -\indexentry{semilinear|hyperpage}{308} -\indexentry{inner product|hyperpage}{308} -\indexentry{pre-Hilbert space|hyperpage}{308} -\indexentry{Cauchy-Schwarz inequality|hyperpage}{308} -\indexentry{Minkowski inequality|hyperpage}{308} -\indexentry{Polarization identity|hyperpage}{308} -\indexentry{orthogonal|hyperpage}{308} -\indexentry{orthogonal complement|hyperpage}{308} -\indexentry{Pythagorean theorem|hyperpage}{308} -\indexentry{Parallelogram law|hyperpage}{308} -\indexentry{Hilbert space|hyperpage}{308} -\indexentry{Hilbert space|hyperpage}{308} -\indexentry{minimizer|hyperpage}{309} -\indexentry{Projection theorem|hyperpage}{309} -\indexentry{orthogonal projection on $F$|hyperpage}{309} -\indexentry{Riesz representation theorem|hyperpage}{310} -\indexentry{adjoint operator|hyperpage}{310} -\indexentry{self-adjoint|hyperpage}{310} -\indexentry{orthogonal system|hyperpage}{310} -\indexentry{orthonormal system|hyperpage}{310} -\indexentry{complete|hyperpage}{310} -\indexentry{Hilbert basis|hyperpage}{310} -\indexentry{Fourier coefficients|hyperpage}{311} -\indexentry{Fourier series|hyperpage}{311} -\indexentry{Gram-Schmidt process|hyperpage}{311} -\indexentry{orthonormalization|hyperpage}{311} -\indexentry{Bessel's inequality|hyperpage}{311} -\indexentry{Fourier transform|hyperpage}{311} -\indexentry{Riesz-Fischer theorem|hyperpage}{311} -\indexentry{Parseval identity|hyperpage}{311} -\indexentry{Spectral theorem|hyperpage}{311} -\indexentry{Hilbert-Schmidt spectral representation theorem|hyperpage}{311} -\indexentry{Fredholm alternative|hyperpage}{312} -\indexentry{Law of total probability|hyperpage}{313} -\indexentry{Substitution principle|hyperpage}{313} -\indexentry{Law of total expectation|hyperpage}{313} -\indexentry{Wald theorem|hyperpage}{313} -\indexentry{probability-generating function|hyperpage}{314} -\indexentry{pgf|hyperpage}{314} -\indexentry{Stochastic process|hyperpage}{314} -\indexentry{stochastic process|hyperpage}{314} -\indexentry{parameter set|hyperpage}{314} -\indexentry{state space|hyperpage}{314} -\indexentry{Gambler's ruin problem|hyperpage}{316} -\indexentry{Markov chain|hyperpage}{316} -\indexentry{Markov property|hyperpage}{316} -\indexentry{time-homogeneous Markov chain|hyperpage}{316} -\indexentry{state space|hyperpage}{316} -\indexentry{states|hyperpage}{316} -\indexentry{Stochastic matrix|hyperpage}{316} -\indexentry{stochastic matrix|hyperpage}{316} -\indexentry{transition probabilities|hyperpage}{316} -\indexentry{transition matrix|hyperpage}{316} -\indexentry{initial distribution|hyperpage}{316} -\indexentry{Random walk|hyperpage}{317} -\indexentry{random walk|hyperpage}{317} -\indexentry{simple random walk|hyperpage}{317} -\indexentry{$n$-step transition probabilities|hyperpage}{317} -\indexentry{$n$-step transition matrix|hyperpage}{317} -\indexentry{$n$-step distribution|hyperpage}{317} -\indexentry{Chapman-Kolmogorov equation|hyperpage}{318} -\indexentry{reachable|hyperpage}{318} -\indexentry{communicate|hyperpage}{318} -\indexentry{irreducible class|hyperpage}{318} -\indexentry{irreducible chain|hyperpage}{318} -\indexentry{period|hyperpage}{318} -\indexentry{aperiodic|hyperpage}{318} -\indexentry{aperiodic|hyperpage}{318} -\indexentry{filtration|hyperpage}{319} -\indexentry{filtration space|hyperpage}{319} -\indexentry{$\sigma $-algebra generated by $\boldsymbol {\mathrm {X}}$|hyperpage}{319} -\indexentry{stopping time|hyperpage}{319} -\indexentry{Strong Markov property|hyperpage}{319} +\indexentry{Lebesgue measure|hyperpage}{292} +\indexentry{real function|hyperpage}{293} +\indexentry{finite|hyperpage}{293} +\indexentry{Lebesgue measurable|hyperpage}{293} +\indexentry{measurable|hyperpage}{293} +\indexentry{Borel $\sigma $-algebra|hyperpage}{294} +\indexentry{Borel measurable|hyperpage}{294} +\indexentry{simple function|hyperpage}{294} +\indexentry{integral of $s$ over $\ensuremath {\mathbb {R}}^n$|hyperpage}{294} +\indexentry{integral of $s$ over a measurable set $E$|hyperpage}{294} +\indexentry{integral of $f$ over $\ensuremath {\mathbb {R}}^n$|hyperpage}{295} +\indexentry{integral of $f$ over a measurable set $E\subseteq \ensuremath {\mathbb {R}}^n$|hyperpage}{295} +\indexentry{Monotone convergence theorem|hyperpage}{295} +\indexentry{Chebyshev's inequality|hyperpage}{296} +\indexentry{Fatou's lemma|hyperpage}{296} +\indexentry{integral of $f$ over $E$|hyperpage}{296} +\indexentry{integrable function over $E$|hyperpage}{296} +\indexentry{Dominated convergence theorem|hyperpage}{296} +\indexentry{converge in mean|hyperpage}{297} +\indexentry{exists and it is finite|hyperpage}{297} +\indexentry{Mean value theorem for integrals|hyperpage}{297} +\indexentry{Barrow's law|hyperpage}{297} +\indexentry{Fundamental theorem of calculus|hyperpage}{297} +\indexentry{Integration by parts|hyperpage}{297} +\indexentry{Hardy-Littlewood maximal function|hyperpage}{297} +\indexentry{Hardy-Littlewood maximal function|hyperpage}{297} +\indexentry{Lebesgue differentiation theorem|hyperpage}{298} +\indexentry{section|hyperpage}{298} +\indexentry{Tonelli's theorem|hyperpage}{298} +\indexentry{Fubini's theorem|hyperpage}{299} +\indexentry{change of variables|hyperpage}{299} +\indexentry{Change of variables|hyperpage}{299} +\indexentry{distance|hyperpage}{299} +\indexentry{triangular inequality|hyperpage}{299} +\indexentry{metric space|hyperpage}{299} +\indexentry{complete|hyperpage}{299} +\indexentry{norm|hyperpage}{299} +\indexentry{triangular inequality|hyperpage}{299} +\indexentry{normed vector space|hyperpage}{299} +\indexentry{convergent series|hyperpage}{299} +\indexentry{absolutely convergent|hyperpage}{299} +\indexentry{Banach space|hyperpage}{300} +\indexentry{Banach space|hyperpage}{300} +\indexentry{uniform norm|hyperpage}{300} +\indexentry{total subset|hyperpage}{300} +\indexentry{separable|hyperpage}{300} +\indexentry{quotient space|hyperpage}{301} +\indexentry{Young's inequality for products|hyperpage}{301} +\indexentry{Hölder conjugates|hyperpage}{301} +\indexentry{Hölder's inequality|hyperpage}{301} +\indexentry{Minkowski inequality|hyperpage}{302} +\indexentry{uniform norm|hyperpage}{302} +\indexentry{uniform norm|hyperpage}{302} +\indexentry{supremum norm|hyperpage}{302} +\indexentry{subalgebra|hyperpage}{302} +\indexentry{separating set|hyperpage}{302} +\indexentry{separate the points|hyperpage}{302} +\indexentry{vanishes nowhere|hyperpage}{302} +\indexentry{self-conjugate|hyperpage}{303} +\indexentry{Stone-Weierstra\ss \ theorem|hyperpage}{303} +\indexentry{pointwise bounded|hyperpage}{303} +\indexentry{locally bounded|hyperpage}{303} +\indexentry{uniformly bounded|hyperpage}{303} +\indexentry{equicontinuous at a point|hyperpage}{303} +\indexentry{pointwise equicontinuous|hyperpage}{303} +\indexentry{uniformly equicontinuous|hyperpage}{303} +\indexentry{Arzelà-Ascoli theorem|hyperpage}{303} +\indexentry{operator|hyperpage}{304} +\indexentry{norm|hyperpage}{304} +\indexentry{sublinear|hyperpage}{305} +\indexentry{Marcinkiewicz interpolation theorem|hyperpage}{305} +\indexentry{topological homeomorphism|hyperpage}{305} +\indexentry{isomorphic|hyperpage}{305} +\indexentry{is finer than|hyperpage}{305} +\indexentry{equivalent|hyperpage}{305} +\indexentry{Almost orthogonality lemma|hyperpage}{305} +\indexentry{Riesz's theorem|hyperpage}{305} +\indexentry{normed algebra|hyperpage}{306} +\indexentry{dual space|hyperpage}{306} +\indexentry{compact operator|hyperpage}{306} +\indexentry{Fredholm operator with kernel $K$|hyperpage}{306} +\indexentry{Volterra operator with kernel $K$|hyperpage}{306} +\indexentry{Hilbert-Schmidt operator with kernel $K$|hyperpage}{306} +\indexentry{finite-rank operator|hyperpage}{306} +\indexentry{Neumann series|hyperpage}{307} +\indexentry{convex functional|hyperpage}{307} +\indexentry{Hahn-Banach theorem|hyperpage}{307} +\indexentry{prolongation|hyperpage}{307} +\indexentry{Seminorm|hyperpage}{307} +\indexentry{seminorm|hyperpage}{307} +\indexentry{Hahn-Banach theorem|hyperpage}{307} +\indexentry{Hahn-Banach theorem|hyperpage}{307} +\indexentry{reflexive|hyperpage}{307} +\indexentry{dual map|hyperpage}{307} +\indexentry{proper subspace of $T$|hyperpage}{308} +\indexentry{eigenvectors|hyperpage}{308} +\indexentry{eigenvalue|hyperpage}{308} +\indexentry{spectrum|hyperpage}{308} +\indexentry{spectral values|hyperpage}{308} +\indexentry{Baire's theorem|hyperpage}{308} +\indexentry{Open mapping theorem|hyperpage}{308} +\indexentry{Closed graph theorem|hyperpage}{308} +\indexentry{Banach-Steinhaus theorem|hyperpage}{308} +\indexentry{semilinear|hyperpage}{309} +\indexentry{inner product|hyperpage}{309} +\indexentry{pre-Hilbert space|hyperpage}{309} +\indexentry{Cauchy-Schwarz inequality|hyperpage}{309} +\indexentry{Minkowski inequality|hyperpage}{309} +\indexentry{Polarization identity|hyperpage}{309} +\indexentry{orthogonal|hyperpage}{309} +\indexentry{orthogonal complement|hyperpage}{309} +\indexentry{Pythagorean theorem|hyperpage}{309} +\indexentry{Parallelogram law|hyperpage}{309} +\indexentry{Hilbert space|hyperpage}{309} +\indexentry{Hilbert space|hyperpage}{309} +\indexentry{minimizer|hyperpage}{310} +\indexentry{Projection theorem|hyperpage}{310} +\indexentry{orthogonal projection on $F$|hyperpage}{310} +\indexentry{Riesz representation theorem|hyperpage}{311} +\indexentry{adjoint operator|hyperpage}{311} +\indexentry{self-adjoint|hyperpage}{311} +\indexentry{orthogonal system|hyperpage}{311} +\indexentry{orthonormal system|hyperpage}{311} +\indexentry{complete|hyperpage}{311} +\indexentry{Hilbert basis|hyperpage}{311} +\indexentry{Fourier coefficients|hyperpage}{312} +\indexentry{Fourier series|hyperpage}{312} +\indexentry{Gram-Schmidt process|hyperpage}{312} +\indexentry{orthonormalization|hyperpage}{312} +\indexentry{Bessel's inequality|hyperpage}{312} +\indexentry{Fourier transform|hyperpage}{312} +\indexentry{Riesz-Fischer theorem|hyperpage}{312} +\indexentry{Parseval identity|hyperpage}{312} +\indexentry{Spectral theorem|hyperpage}{312} +\indexentry{Hilbert-Schmidt spectral representation theorem|hyperpage}{312} +\indexentry{Fredholm alternative|hyperpage}{313} +\indexentry{Law of total probability|hyperpage}{314} +\indexentry{Substitution principle|hyperpage}{314} +\indexentry{Law of total expectation|hyperpage}{314} +\indexentry{Wald theorem|hyperpage}{314} +\indexentry{probability-generating function|hyperpage}{315} +\indexentry{pgf|hyperpage}{315} +\indexentry{Stochastic process|hyperpage}{315} +\indexentry{stochastic process|hyperpage}{315} +\indexentry{parameter set|hyperpage}{315} +\indexentry{state space|hyperpage}{315} +\indexentry{independent|hyperpage}{315} +\indexentry{Gambler's ruin problem|hyperpage}{317} +\indexentry{Markov chain|hyperpage}{317} +\indexentry{Markov property|hyperpage}{317} +\indexentry{time-homogeneous Markov chain|hyperpage}{317} +\indexentry{state space|hyperpage}{317} +\indexentry{states|hyperpage}{317} +\indexentry{Stochastic matrix|hyperpage}{317} +\indexentry{stochastic matrix|hyperpage}{317} +\indexentry{transition probabilities|hyperpage}{317} +\indexentry{transition matrix|hyperpage}{317} +\indexentry{initial distribution|hyperpage}{317} +\indexentry{Random walk|hyperpage}{318} +\indexentry{random walk|hyperpage}{318} +\indexentry{simple random walk|hyperpage}{318} +\indexentry{$n$-step transition probabilities|hyperpage}{318} +\indexentry{$n$-step transition matrix|hyperpage}{318} +\indexentry{$n$-step distribution|hyperpage}{318} +\indexentry{Chapman-Kolmogorov equation|hyperpage}{319} +\indexentry{reachable|hyperpage}{319} +\indexentry{communicate|hyperpage}{319} +\indexentry{irreducible class|hyperpage}{319} +\indexentry{irreducible chain|hyperpage}{319} +\indexentry{period|hyperpage}{319} +\indexentry{aperiodic|hyperpage}{319} +\indexentry{aperiodic|hyperpage}{319} +\indexentry{filtration|hyperpage}{320} +\indexentry{filtration space|hyperpage}{320} +\indexentry{$\sigma $-algebra generated by $\boldsymbol {\mathrm {X}}$|hyperpage}{320} +\indexentry{stopping time|hyperpage}{320} \indexentry{Strong Markov property|hyperpage}{320} -\indexentry{transient|hyperpage}{320} -\indexentry{recurrent|hyperpage}{320} -\indexentry{$k$-th hitting time|hyperpage}{320} -\indexentry{recurrent|hyperpage}{321} +\indexentry{Strong Markov property|hyperpage}{321} \indexentry{transient|hyperpage}{321} -\indexentry{Polya's theorem on $\ensuremath {\mathbb {Z}}$|hyperpage}{321} -\indexentry{Polya's theorem on $\ensuremath {\mathbb {Z}}^2$|hyperpage}{322} -\indexentry{Polya's theorem on $\ensuremath {\mathbb {Z}}^3$|hyperpage}{322} -\indexentry{positive recurrent|hyperpage}{322} -\indexentry{null recurrent|hyperpage}{322} -\indexentry{Ergotic theorem|hyperpage}{322} -\indexentry{Erogotic theorem|hyperpage}{322} -\indexentry{stationary distribution|hyperpage}{323} -\indexentry{continuous-time Markov chain|hyperpage}{323} -\indexentry{transition probabilities|hyperpage}{323} -\indexentry{Chapman-Kolmogorov equation|hyperpage}{323} -\indexentry{Poisson process|hyperpage}{324} -\indexentry{independent increments|hyperpage}{324} -\indexentry{stationary increments|hyperpage}{324} -\indexentry{trajectories|hyperpage}{324} -\indexentry{càd|hyperpage}{324} -\indexentry{holding times|hyperpage}{324} -\indexentry{inter-arrival times|hyperpage}{324} -\indexentry{infinitesimal generator|hyperpage}{325} -\indexentry{infinitesimal transition scheme|hyperpage}{325} -\indexentry{Kolmogorov's backward equation|hyperpage}{325} -\indexentry{Kolmogorov's forward equation|hyperpage}{325} -\indexentry{jump process|hyperpage}{326} -\indexentry{regular|hyperpage}{326} -\indexentry{stable|hyperpage}{326} -\indexentry{conservative|hyperpage}{326} -\indexentry{stationary distribution|hyperpage}{326} -\indexentry{balance equation|hyperpage}{326} -\indexentry{irreducible|hyperpage}{326} -\indexentry{limit distribution|hyperpage}{326} -\indexentry{birth and death processes|hyperpage}{327} -\indexentry{pure death process|hyperpage}{327} -\indexentry{pure birth process|hyperpage}{327} -\indexentry{gaussian vector|hyperpage}{327} -\indexentry{gaussian process|hyperpage}{327} -\indexentry{mean function|hyperpage}{327} -\indexentry{covariance function|hyperpage}{327} -\indexentry{Brownian motion|hyperpage}{327} -\indexentry{standard|hyperpage}{327} -\indexentry{stochastically equivalent|hyperpage}{328} -\indexentry{version|hyperpage}{328} -\indexentry{indistinguishable|hyperpage}{328} -\indexentry{Kolmogorov's continuity theorem|hyperpage}{328} -\indexentry{Paley-Wiener-Zygmund theorem|hyperpage}{329} -\indexentry{Law of the iterated logarithm|hyperpage}{330} -\indexentry{Finite-dimensional distributions|hyperpage}{330} -\indexentry{finite-dimensional distributions|hyperpage}{330} -\indexentry{consistency condition|hyperpage}{330} -\indexentry{Kolmogorov extension theorem|hyperpage}{330} +\indexentry{recurrent|hyperpage}{321} +\indexentry{$k$-th hitting time|hyperpage}{321} +\indexentry{recurrent|hyperpage}{323} +\indexentry{transient|hyperpage}{323} +\indexentry{Polya's theorem on $\ensuremath {\mathbb {Z}}$|hyperpage}{323} +\indexentry{Polya's theorem on $\ensuremath {\mathbb {Z}}^2$|hyperpage}{323} +\indexentry{Polya's theorem on $\ensuremath {\mathbb {Z}}^3$|hyperpage}{323} +\indexentry{positive recurrent|hyperpage}{323} +\indexentry{null recurrent|hyperpage}{323} +\indexentry{Ergotic theorem|hyperpage}{323} +\indexentry{Ergotic theorem|hyperpage}{324} +\indexentry{stationary distribution|hyperpage}{324} +\indexentry{continuous-time Markov chain|hyperpage}{325} +\indexentry{transition probabilities|hyperpage}{325} +\indexentry{Chapman-Kolmogorov equation|hyperpage}{325} +\indexentry{Poisson process|hyperpage}{325} +\indexentry{independent increments|hyperpage}{325} +\indexentry{stationary increments|hyperpage}{325} +\indexentry{trajectories|hyperpage}{325} +\indexentry{càd|hyperpage}{325} +\indexentry{holding times|hyperpage}{326} +\indexentry{inter-arrival times|hyperpage}{326} +\indexentry{infinitesimal generator|hyperpage}{327} +\indexentry{infinitesimal transition scheme|hyperpage}{327} +\indexentry{Kolmogorov's backward equation|hyperpage}{327} +\indexentry{Kolmogorov's forward equation|hyperpage}{327} +\indexentry{jump process|hyperpage}{327} +\indexentry{regular|hyperpage}{327} +\indexentry{stable|hyperpage}{327} +\indexentry{conservative|hyperpage}{327} +\indexentry{stationary distribution|hyperpage}{328} +\indexentry{balance equation|hyperpage}{328} +\indexentry{irreducible|hyperpage}{328} +\indexentry{limit distribution|hyperpage}{328} +\indexentry{birth and death process|hyperpage}{328} +\indexentry{pure death process|hyperpage}{328} +\indexentry{pure birth process|hyperpage}{328} +\indexentry{Reuter criterion|hyperpage}{329} +\indexentry{gaussian vector|hyperpage}{329} +\indexentry{gaussian process|hyperpage}{329} +\indexentry{mean function|hyperpage}{329} +\indexentry{covariance function|hyperpage}{329} +\indexentry{Brownian motion|hyperpage}{329} +\indexentry{Wiener process|hyperpage}{329} +\indexentry{standard|hyperpage}{329} +\indexentry{stochastically equivalent|hyperpage}{330} +\indexentry{version|hyperpage}{330} +\indexentry{indistinguishable|hyperpage}{330} +\indexentry{Kolmogorov's continuity theorem|hyperpage}{330} +\indexentry{Paley-Wiener-Zygmund theorem|hyperpage}{330} +\indexentry{$d$-dimensional standard Brownian motion|hyperpage}{331} +\indexentry{Law of the iterated logarithm|hyperpage}{332} +\indexentry{Finite-dimensional distributions|hyperpage}{332} +\indexentry{finite-dimensional distributions|hyperpage}{332} +\indexentry{consistency condition|hyperpage}{332} +\indexentry{Kolmogorov extension theorem|hyperpage}{332} diff --git a/main_math.ilg b/main_math.ilg index 4380993..1e7cd1c 100644 --- a/main_math.ilg +++ b/main_math.ilg @@ -1,6 +1,6 @@ This is makeindex, version 2.17 [TeX Live 2023] (kpathsea + Thai support). -Scanning input file main_math.idx.......done (3045 entries accepted, 0 rejected). -Sorting entries.............................done (38451 comparisons). -Generating output file main_math.ind......done (2601 lines written, 0 warnings). +Scanning input file main_math.idx.......done (3055 entries accepted, 0 rejected). +Sorting entries..............................done (39602 comparisons). +Generating output file main_math.ind......done (2608 lines written, 0 warnings). Output written in main_math.ind. Transcript written in main_math.ilg. diff --git a/main_math.ind b/main_math.ind index d5ebb90..e5869d4 100644 --- a/main_math.ind +++ b/main_math.ind @@ -24,24 +24,25 @@ \item $\ensuremath {\mathbb {R}}$-differentiable, \hyperpage{109} \item $\omega $-limit point, \hyperpage{130} \item $\omega $-limit set, \hyperpage{130} - \item $\sigma $-additivity, \hyperpage{172}, \hyperpage{289} - \item $\sigma $-algebra, \hyperpage{171}, \hyperpage{289} + \item $\sigma $-additivity, \hyperpage{172}, \hyperpage{290} + \item $\sigma $-algebra, \hyperpage{171}, \hyperpage{290} \item $\sigma $-algebra generated, \hyperpage{171} \item $\sigma $-algebra generated by $\boldsymbol {\mathrm {X}}$, - \hyperpage{319} + \hyperpage{320} \item $\sigma $-algebra of all Lebesgue measurable sets in $\ensuremath {\mathbb {R}}^n$, \hyperpage{174} + \item $d$-dimensional standard Brownian motion, \hyperpage{331} \item $i$-th pivot, \hyperpage{13} \item $k$-linear map, \hyperpage{154} - \item $k$-th hitting time, \hyperpage{320} + \item $k$-th hitting time, \hyperpage{321} \item $m$-dimensional projective subvariety, \hyperpage{66} - \item $m$-th dyadic cube, \hyperpage{289} + \item $m$-th dyadic cube, \hyperpage{290} \item $n$-dimensional projective space, \hyperpage{66} \item $n$-dimensional volume, \hyperpage{57, 58} \item $n$-periodic, \hyperpage{89} - \item $n$-step distribution, \hyperpage{317} - \item $n$-step transition matrix, \hyperpage{317} - \item $n$-step transition probabilities, \hyperpage{317} + \item $n$-step distribution, \hyperpage{318} + \item $n$-step transition matrix, \hyperpage{318} + \item $n$-step transition probabilities, \hyperpage{318} \item $n$-th Fourier coefficient, \hyperpage{82} \item $n$-th Lyapunov constant, \hyperpage{225} \item $n$-th ball, \hyperpage{209} @@ -68,7 +69,7 @@ \indexspace \item A-stable, \hyperpage{260} - \item a.e., \hyperpage{291} + \item a.e., \hyperpage{292} \item Abel's summation formula, \hyperpage{76}, \hyperpage{105} \item Abel's test, \hyperpage{76, 77}, \hyperpage{79}, \hyperpage{106} @@ -81,12 +82,12 @@ \item absolute value, \hyperpage{24} \item absolutely continuous, \hyperpage{177}, \hyperpage{179} \item absolutely convergent, \hyperpage{76}, \hyperpage{105}, - \hyperpage{116}, \hyperpage{298} + \hyperpage{116}, \hyperpage{299} \item absolutely stable, \hyperpage{260} \item acceptation region, \hyperpage{198} \item ACCP, \hyperpage{45} \item accumulation point, \hyperpage{25}, \hyperpage{53} - \item action, \hyperpage{210}, \hyperpage{276} + \item action, \hyperpage{210}, \hyperpage{277} \item Adams method, \hyperpage{261} \item Adams-Bashforth method, \hyperpage{261} \item Adams-Moulton method, \hyperpage{261} @@ -95,7 +96,7 @@ \item adjacency matrix, \hyperpage{49} \item adjacent, \hyperpage{48} \item adjoint, \hyperpage{23} - \item adjoint operator, \hyperpage{310} + \item adjoint operator, \hyperpage{311} \item adjugate matrix, \hyperpage{14} \item adjusted coefficient of determination, \hyperpage{251} \item affine combination, \hyperpage{68} @@ -118,16 +119,16 @@ \item algebraic value, \hyperpage{152} \item algebraically closed, \hyperpage{169} \item algebraically closed field, \hyperpage{20} - \item almost everywhere, \hyperpage{291} - \item Almost orthogonality lemma, \hyperpage{304} + \item almost everywhere, \hyperpage{292} + \item Almost orthogonality lemma, \hyperpage{305} \item alternating, \hyperpage{154} \item alternating group, \hyperpage{41} \item alternating series, \hyperpage{76} \item alternative hypothesis, \hyperpage{198} \item amplification factor, \hyperpage{269} \item amplification polynomial, \hyperpage{272} - \item Ampère's law, \hyperpage{276} - \item Ampère-Maxwell circuital law, \hyperpage{276} + \item Ampère's law, \hyperpage{277} + \item Ampère-Maxwell circuital law, \hyperpage{277} \item an asymptotically normal estimator, \hyperpage{193} \item analytic, \hyperpage{78}, \hyperpage{112} \item analytic at $a$, \hyperpage{31} @@ -136,7 +137,7 @@ \item angle-preserving, \hyperpage{148} \item anisotropic, \hyperpage{73} \item annihilator, \hyperpage{18} - \item aperiodic, \hyperpage{318} + \item aperiodic, \hyperpage{319} \item approximation of identity, \hyperpage{78}, \hyperpage{240} \item approximations of the identity, \hyperpage{234} \item arc length, \hyperpage{59} @@ -149,7 +150,7 @@ \item Argument principle, \hyperpage{117} \item Artin's lemma, \hyperpage{168} \item Arzelà-Ascoli theorem, \hyperpage{120}, \hyperpage{124}, - \hyperpage{302} + \hyperpage{303} \item ASA criterion, \hyperpage{63} \item ascending chain condition on principal ideals, \hyperpage{45} \item associated, \hyperpage{44} @@ -168,7 +169,7 @@ \item attractor fixed point, \hyperpage{90} \item auto-adjoint, \hyperpage{23} \item autonomous, \hyperpage{121} - \item average, \hyperpage{287} + \item average, \hyperpage{288} \item average response, \hyperpage{252} \item Axiom of Archimedes, \hyperpage{62} \item Axiom of choice, \hyperpage{212} @@ -181,15 +182,15 @@ \indexspace \item backward Euler method, \hyperpage{257} - \item Backward-time central-space, \hyperpage{272} - \item Baire's theorem, \hyperpage{307} - \item balance equation, \hyperpage{326} + \item Backward-time central-space, \hyperpage{267}, \hyperpage{273} + \item Baire's theorem, \hyperpage{308} + \item balance equation, \hyperpage{328} \item ball, \hyperpage{204} \item Banach fixed-point theorem, \hyperpage{124} - \item Banach space, \hyperpage{299} - \item Banach-Steinhaus theorem, \hyperpage{307} + \item Banach space, \hyperpage{300} + \item Banach-Steinhaus theorem, \hyperpage{308} \item bandlimited, \hyperpage{238} - \item Barrow's law, \hyperpage{296} + \item Barrow's law, \hyperpage{297} \item barycenter, \hyperpage{68} \item Basic bootstrap confidence interval, \hyperpage{201} \item basic bootstrap confidence interval, \hyperpage{201} @@ -206,7 +207,7 @@ \item Bernoulli distribution, \hyperpage{176} \item Bernoulli numbers, \hyperpage{95} \item Bernoulli polynomials, \hyperpage{95} - \item Bessel's inequality, \hyperpage{86}, \hyperpage{311} + \item Bessel's inequality, \hyperpage{86}, \hyperpage{312} \item Beta distribution, \hyperpage{178} \item beta distribution, \hyperpage{178} \item beta function, \hyperpage{178} @@ -225,7 +226,7 @@ \item Binomial theorem, \hyperpage{118} \item binormal vector, \hyperpage{142} \item bipartite, \hyperpage{49} - \item birth and death processes, \hyperpage{327} + \item birth and death process, \hyperpage{328} \item Bisection method, \hyperpage{89} \item block matrix, \hyperpage{14} \item blow-down, \hyperpage{224} @@ -243,8 +244,8 @@ \item bootstrap distribution, \hyperpage{201} \item Bootstrap-t confidence interval, \hyperpage{201} \item bootstrap-t confidence interval, \hyperpage{201} - \item Borel $\sigma $-algebra, \hyperpage{171}, \hyperpage{293} - \item Borel measurable, \hyperpage{180}, \hyperpage{293} + \item Borel $\sigma $-algebra, \hyperpage{171}, \hyperpage{294} + \item Borel measurable, \hyperpage{180}, \hyperpage{294} \item Borel's theorem, \hyperpage{189} \item Borsuk-Ulam theorem, \hyperpage{214} \item Boundary, \hyperpage{206} @@ -259,9 +260,10 @@ \item Box-Cox transformation, \hyperpage{256} \item bridge, \hyperpage{49} \item Brouwer's fixed-point theorem, \hyperpage{214} - \item Brownian motion, \hyperpage{327} + \item Brownian motion, \hyperpage{329} \item Broyden's method, \hyperpage{262} \item Broyden-Fletcher-Goldfarb-Shanno method, \hyperpage{263} + \item BTCS, \hyperpage{267} \item Budan-Fourier theorem, \hyperpage{91} \item bump functions, \hyperpage{239} \item Butcher tableau, \hyperpage{259} @@ -290,7 +292,7 @@ \item Cauchy distribution, \hyperpage{178} \item Cauchy in $p$-th mean, \hyperpage{188} \item Cauchy in probability, \hyperpage{186} - \item Cauchy momentum equation, \hyperpage{275} + \item Cauchy momentum equation, \hyperpage{276} \item Cauchy principal value, \hyperpage{240} \item Cauchy problem, \hyperpage{121} \item Cauchy product, \hyperpage{105} @@ -306,7 +308,7 @@ \item Cauchy-Riemann equations, \hyperpage{110} \item Cauchy-Riemann theorem, \hyperpage{110} \item Cauchy-Schwarz inequality, \hyperpage{23}, \hyperpage{52}, - \hyperpage{81}, \hyperpage{184}, \hyperpage{308} + \hyperpage{81}, \hyperpage{184}, \hyperpage{309} \item Cavalieri's principle, \hyperpage{58} \item Cayley's theorem, \hyperpage{38} \item Cayley-Hamilton theorem, \hyperpage{20} @@ -329,15 +331,15 @@ \item Change of basis formula, \hyperpage{17} \item change of parameter, \hyperpage{140} \item Change of variable theorem, \hyperpage{59} - \item Change of variables, \hyperpage{158}, \hyperpage{298} - \item change of variables, \hyperpage{298} + \item Change of variables, \hyperpage{158}, \hyperpage{299} + \item change of variables, \hyperpage{299} \item change-of-basis matrix, \hyperpage{17} \item Chaos, \hyperpage{229} \item chaotic, \hyperpage{229} \item chaotic behavior, \hyperpage{90} - \item Chapman-Kolmogorov equation, \hyperpage{318}, \hyperpage{323} + \item Chapman-Kolmogorov equation, \hyperpage{319}, \hyperpage{325} \item characteristic, \hyperpage{161} - \item characteristic curves, \hyperpage{277} + \item characteristic curves, \hyperpage{278} \item Characteristic equation, \hyperpage{122}, \hyperpage{221} \item characteristic equation, \hyperpage{122} \item Characteristic function, \hyperpage{190} @@ -345,7 +347,7 @@ \item characteristic polynomial, \hyperpage{19}, \hyperpage{48} \item Chebyshev method, \hyperpage{89} \item Chebyshev polynomials, \hyperpage{95} - \item Chebyshev's inequality, \hyperpage{185}, \hyperpage{295} + \item Chebyshev's inequality, \hyperpage{185}, \hyperpage{296} \item chi-squared distribution with $n$ degrees of freedom, \hyperpage{196} \item Chinese remainder theorem, \hyperpage{10} @@ -381,7 +383,7 @@ \item closed ball, \hyperpage{53} \item Closed forms, \hyperpage{47} \item Closed function, \hyperpage{207} - \item Closed graph theorem, \hyperpage{307} + \item Closed graph theorem, \hyperpage{308} \item Closed interval, \hyperpage{24} \item Closed Newton-Cotes Formulas, \hyperpage{94} \item closed trail, \hyperpage{48} @@ -400,10 +402,10 @@ \item column rank, \hyperpage{15} \item Combinations with repetition, \hyperpage{9} \item Combinations without repetition, \hyperpage{9} - \item communicate, \hyperpage{318} + \item communicate, \hyperpage{319} \item commutative, \hyperpage{36} \item compact, \hyperpage{53}, \hyperpage{212} - \item compact operator, \hyperpage{305} + \item compact operator, \hyperpage{306} \item Compact space, \hyperpage{212} \item compact subset, \hyperpage{212} \item compact support, \hyperpage{78} @@ -416,7 +418,7 @@ \item complement, \hyperpage{5} \item Complementary property, \hyperpage{51} \item complementary subspace, \hyperpage{16} - \item complete, \hyperpage{53}, \hyperpage{298}, \hyperpage{310} + \item complete, \hyperpage{53}, \hyperpage{299}, \hyperpage{311} \item complete graph, \hyperpage{48} \item Complete pivoting, \hyperpage{101} \item complex conjugate, \hyperpage{23}, \hyperpage{103} @@ -479,9 +481,9 @@ \item Connected sum, \hyperpage{216} \item connected sum, \hyperpage{216} \item Connected sum of surfaces, \hyperpage{216} - \item Conservation of energy, \hyperpage{280} - \item conservative, \hyperpage{60}, \hyperpage{326} - \item consistency condition, \hyperpage{330} + \item Conservation of energy, \hyperpage{281} + \item conservative, \hyperpage{60}, \hyperpage{327} + \item consistency condition, \hyperpage{332} \item consistent, \hyperpage{258}, \hyperpage{261}, \hyperpage{268} \item consistent estimator in $L^2$, \hyperpage{193} \item constant coefficients, \hyperpage{121}, \hyperpage{125} @@ -499,22 +501,22 @@ \item continuous, \hyperpage{27}, \hyperpage{54}, \hyperpage{105}, \hyperpage{129} \item continuous at $x_0$, \hyperpage{27} - \item Continuous equation, \hyperpage{275} - \item continuous equation, \hyperpage{275} + \item Continuous equation, \hyperpage{276} + \item continuous equation, \hyperpage{276} \item Continuous function, \hyperpage{207} \item Continuous memorylessness property, \hyperpage{177} \item Continuous uniform distribution, \hyperpage{177} \item continuous uniform distribution, \hyperpage{177} - \item continuous-time Markov chain, \hyperpage{323} + \item continuous-time Markov chain, \hyperpage{325} \item contractible, \hyperpage{138} \item contraction, \hyperpage{54}, \hyperpage{89} \item contrast matrix, \hyperpage{200} - \item converge in mean, \hyperpage{296} + \item converge in mean, \hyperpage{297} \item converge in norm $L^p$, \hyperpage{86} \item convergent, \hyperpage{25}, \hyperpage{53}, \hyperpage{75}, \hyperpage{78}, \hyperpage{97}, \hyperpage{104, 105}, \hyperpage{116}, \hyperpage{258} - \item convergent series, \hyperpage{298} + \item convergent series, \hyperpage{299} \item converges, \hyperpage{104} \item converges absolutely, \hyperpage{79} \item converges almost surely, \hyperpage{187} @@ -524,7 +526,7 @@ \item converges pointwise, \hyperpage{76, 77}, \hyperpage{105, 106} \item converges uniformly, \hyperpage{76, 77}, \hyperpage{105, 106} \item convex, \hyperpage{29}, \hyperpage{114} - \item convex functional, \hyperpage{306} + \item convex functional, \hyperpage{307} \item convolution, \hyperpage{78}, \hyperpage{239}, \hyperpage{243} \item Cook's distance, \hyperpage{254} \item coordinate chart, \hyperpage{146}, \hyperpage{215} @@ -536,18 +538,18 @@ \item Correspondence theorem, \hyperpage{39}, \hyperpage{43} \item countable, \hyperpage{24}, \hyperpage{212} \item Countable subadditivity, \hyperpage{172} - \item Courant-Friedrichs-Lewy condition, \hyperpage{268} + \item Courant-Friedrichs-Lewy condition, \hyperpage{269} \item Covariance, \hyperpage{184} \item covariance, \hyperpage{184}, \hyperpage{248} - \item covariance function, \hyperpage{327} + \item covariance function, \hyperpage{329} \item covariance matrix, \hyperpage{180}, \hyperpage{195}, \hyperpage{248} \item covariant derivative, \hyperpage{151} \item Cover, \hyperpage{212} \item cover, \hyperpage{212} \item Cramér-Rao bound, \hyperpage{195} - \item Crank-Nicolson, \hyperpage{270} - \item Crank-Nicolson scheme, \hyperpage{270}, \hyperpage{272} + \item Crank-Nicolson, \hyperpage{271} + \item Crank-Nicolson scheme, \hyperpage{271}, \hyperpage{273} \item credible interval, \hyperpage{202} \item credible region, \hyperpage{202} \item critical point, \hyperpage{56}, \hyperpage{130} @@ -570,12 +572,12 @@ \item cyclic, \hyperpage{169} \item cyclic group, \hyperpage{36} \item cyclotomic, \hyperpage{169} - \item càd, \hyperpage{324} + \item càd, \hyperpage{325} \item càdlàg, \hyperpage{175} \indexspace - \item D'Alembert formula, \hyperpage{139}, \hyperpage{279} + \item D'Alembert formula, \hyperpage{139}, \hyperpage{280} \item D'Alembert theorem, \hyperpage{11} \item Dahlquist's theorem, \hyperpage{261} \item damping parameter, \hyperpage{262} @@ -598,7 +600,7 @@ \item dehomogenization, \hyperpage{69} \item Delta method, \hyperpage{195} \item dense, \hyperpage{206} - \item density, \hyperpage{276} + \item density, \hyperpage{277} \item density function, \hyperpage{177} \item Dependence on $\boldsymbol {\mathrm {\lambda }}$, \hyperpage{128} @@ -640,11 +642,11 @@ \item differential 1-form, \hyperpage{61} \item differential operator over distributions, \hyperpage{244} \item differential system, \hyperpage{121} - \item differentiation operator, \hyperpage{283} - \item diffusion coefficient, \hyperpage{276} - \item Diffusion equation, \hyperpage{276} - \item diffusion flux, \hyperpage{276} - \item diffusivity, \hyperpage{276} + \item differentiation operator, \hyperpage{284} + \item diffusion coefficient, \hyperpage{277} + \item Diffusion equation, \hyperpage{277} + \item diffusion flux, \hyperpage{277} + \item diffusivity, \hyperpage{277} \item digital topology, \hyperpage{205} \item dihedral group, \hyperpage{41} \item dilatation, \hyperpage{118} @@ -652,7 +654,7 @@ \hyperpage{192} \item Dini's theorem, \hyperpage{84}, \hyperpage{233} \item Dirac's $\delta $ distribution, \hyperpage{240}, - \hyperpage{283} + \hyperpage{284} \item direct, \hyperpage{15} \item direct product, \hyperpage{243} \item Direct sum, \hyperpage{15} @@ -660,9 +662,9 @@ \item director subspace, \hyperpage{69} \item Dirichlet, \hyperpage{273} \item Dirichlet kernel, \hyperpage{83}, \hyperpage{232} - \item Dirichlet problem, \hyperpage{139}, \hyperpage{286} - \item Dirichlet problem in the disc, \hyperpage{286} - \item Dirichlet's principle, \hyperpage{286} + \item Dirichlet problem, \hyperpage{139}, \hyperpage{287} + \item Dirichlet problem in the disc, \hyperpage{287} + \item Dirichlet's principle, \hyperpage{287} \item Dirichlet's test, \hyperpage{76, 77}, \hyperpage{79}, \hyperpage{106} \item disconnected, \hyperpage{213} @@ -681,18 +683,18 @@ \item Discrete uniform distribution, \hyperpage{176} \item discrete uniform distribution, \hyperpage{176} \item discriminant, \hyperpage{168} - \item displacement current, \hyperpage{276} + \item displacement current, \hyperpage{277} \item distance, \hyperpage{52}, \hyperpage{81}, \hyperpage{204}, - \hyperpage{298} + \hyperpage{299} \item distance between two affine subvarieties, \hyperpage{71} \item distance between two points, \hyperpage{71} - \item Distribution, \hyperpage{239}, \hyperpage{283} + \item Distribution, \hyperpage{239}, \hyperpage{284} \item distribution, \hyperpage{175}, \hyperpage{179}, \hyperpage{239}, - \hyperpage{283} + \hyperpage{284} \item distribution functions, \hyperpage{175} \item distribution of order $N\in \ensuremath {\mathbb {N}}\cup \{0\}$, \hyperpage{240} - \item distributional derivative, \hyperpage{241}, \hyperpage{283} + \item distributional derivative, \hyperpage{241}, \hyperpage{284} \item divergence, \hyperpage{60} \item Divergence theorem, \hyperpage{159} \item Divergence theorem on $\ensuremath {\mathbb {R}}^2$, @@ -705,20 +707,20 @@ \item dominant eigenvalue, \hyperpage{99} \item dominant eigenvector, \hyperpage{99} \item Dominated convergence theorem, \hyperpage{183}, \hyperpage{188}, - \hyperpage{295} + \hyperpage{296} \item dot product, \hyperpage{52} \item double, \hyperpage{89} \item Double dual space, \hyperpage{18} \item double dual space, \hyperpage{18} \item Du Bois-Reymond's test, \hyperpage{106} - \item Du-Fort-Frankel scheme, \hyperpage{272} + \item Du-Fort-Frankel scheme, \hyperpage{273} \item dual basis, \hyperpage{18} \item Dual map, \hyperpage{18} - \item dual map, \hyperpage{18}, \hyperpage{306} + \item dual map, \hyperpage{18}, \hyperpage{307} \item dual program, \hyperpage{50} - \item dual space, \hyperpage{18}, \hyperpage{305} + \item dual space, \hyperpage{18}, \hyperpage{306} \item Duality principle, \hyperpage{65} - \item Duhamel principle, \hyperpage{285} + \item Duhamel principle, \hyperpage{286} \item dummy variable, \hyperpage{252} \item dyadic partition of order $n$, \hyperpage{182} \item dynamical system, \hyperpage{129} @@ -728,27 +730,27 @@ \item edges, \hyperpage{159} \item efficient estimator, \hyperpage{195} \item eigenspace, \hyperpage{19} - \item eigenvalue, \hyperpage{307} + \item eigenvalue, \hyperpage{308} \item eigenvector, \hyperpage{19} - \item eigenvectors, \hyperpage{307} - \item Eikonal equation, \hyperpage{276} + \item eigenvectors, \hyperpage{308} + \item Eikonal equation, \hyperpage{277} \item Eisenstein's criterion, \hyperpage{44}, \hyperpage{46} \item either $P$ or $Q$ are true, \hyperpage{7} - \item elastodynamics, \hyperpage{275} + \item elastodynamics, \hyperpage{276} \item elementary region, \hyperpage{58} \item elementary symmetric polynomials, \hyperpage{162} \item elliptic, \hyperpage{73}, \hyperpage{272} - \item elliptic equation, \hyperpage{275} + \item elliptic equation, \hyperpage{276} \item elliptic point, \hyperpage{149} \item elliptic sector, \hyperpage{132} \item embedded methods, \hyperpage{259} \item empirical distribution, \hyperpage{201} \item empty set, \hyperpage{5} - \item energy functional, \hyperpage{286} + \item energy functional, \hyperpage{287} \item entire function, \hyperpage{108} \item Entropy, \hyperpage{255} \item entropy, \hyperpage{255} - \item Entropy condition, \hyperpage{278} + \item Entropy condition, \hyperpage{279} \item envelope, \hyperpage{143} \item equal, \hyperpage{5} \item equal almost surely, \hyperpage{175} @@ -756,7 +758,7 @@ \item equally-spaced, \hyperpage{92} \item equation of the hyperplane, \hyperpage{66} \item equicontinuous, \hyperpage{120} - \item equicontinuous at a point, \hyperpage{124}, \hyperpage{302} + \item equicontinuous at a point, \hyperpage{124}, \hyperpage{303} \item equilibrium point, \hyperpage{220} \item equiprobable space, \hyperpage{176} \item equivalence, \hyperpage{130} @@ -764,10 +766,9 @@ \item equivalence relation, \hyperpage{7} \item equivalent, \hyperpage{12}, \hyperpage{16}, \hyperpage{22}, \hyperpage{72}, \hyperpage{129}, \hyperpage{131}, - \hyperpage{304} + \hyperpage{305} \item equivalent dynamical systems, \hyperpage{130} - \item Ergotic theorem, \hyperpage{322} - \item Erogotic theorem, \hyperpage{322} + \item Ergotic theorem, \hyperpage{323, 324} \item error of type I, \hyperpage{198} \item error of type II, \hyperpage{198} \item error sum of squares, \hyperpage{251} @@ -798,7 +799,7 @@ \item Eulerian trail, \hyperpage{49} \item evaluation, \hyperpage{11}, \hyperpage{161} \item even extension, \hyperpage{83} - \item even periodic extension, \hyperpage{281} + \item even periodic extension, \hyperpage{282} \item event, \hyperpage{172} \item evolute, \hyperpage{143} \item exact, \hyperpage{155} @@ -806,7 +807,7 @@ \item Excluded point topology, \hyperpage{205} \item Existence of orthogonal polynomials, \hyperpage{95} \item Existence of the splitting field, \hyperpage{166} - \item exists and it is finite, \hyperpage{296} + \item exists and it is finite, \hyperpage{297} \item Expansive fixed point theorem, \hyperpage{228} \item Expectation, \hyperpage{182} \item expectation, \hyperpage{177}, \hyperpage{181--183}, @@ -814,7 +815,7 @@ \item explicit, \hyperpage{257} \item explicit Euler method, \hyperpage{257} \item explicit form, \hyperpage{121} - \item Explicit scheme in finite differences, \hyperpage{284} + \item Explicit scheme in finite differences, \hyperpage{285} \item exponent, \hyperpage{88} \item exponential, \hyperpage{256} \item Exponential distribution, \hyperpage{177} @@ -841,18 +842,19 @@ \item false, \hyperpage{176} \item Fano configuration, \hyperpage{66} \item fast Fourier transform, \hyperpage{239} - \item Fatou's lemma, \hyperpage{183}, \hyperpage{295} + \item Fatou's lemma, \hyperpage{183}, \hyperpage{296} \item feasible region, \hyperpage{50} \item feasible solution, \hyperpage{50} \item Fejér kernel, \hyperpage{84}, \hyperpage{233} \item Fejér mean, \hyperpage{233} \item Fejér means, \hyperpage{85} \item Fejér's theorem, \hyperpage{85} + \item FEM, \hyperpage{274} \item Fermat's little theorem, \hyperpage{10} - \item Fermat's principle, \hyperpage{276} + \item Fermat's principle, \hyperpage{277} \item FFT, \hyperpage{239} - \item Fick's law, \hyperpage{276} - \item Fick's law of diffusion, \hyperpage{276} + \item Fick's law, \hyperpage{277} + \item Fick's law of diffusion, \hyperpage{277} \item Field, \hyperpage{41} \item field, \hyperpage{41} \item field extension, \hyperpage{163} @@ -860,24 +862,24 @@ \item field of complex numbers, \hyperpage{103} \item field of fractions, \hyperpage{45}, \hyperpage{162} \item field of rational functions, \hyperpage{162} - \item filtration, \hyperpage{319} - \item filtration space, \hyperpage{319} + \item filtration, \hyperpage{320} + \item filtration space, \hyperpage{320} \item finer, \hyperpage{204} \item finer than, \hyperpage{31}, \hyperpage{57} \item finite, \hyperpage{24}, \hyperpage{48}, \hyperpage{163}, - \hyperpage{212}, \hyperpage{292} + \hyperpage{212}, \hyperpage{293} \item finite $k$-th moment, \hyperpage{184} \item finite difference method, \hyperpage{264} \item finite difference scheme, \hyperpage{267} - \item finite element method, \hyperpage{273, 274} + \item finite element method, \hyperpage{274} \item finite expectation, \hyperpage{181--183} \item Finite field, \hyperpage{165} \item finite field, \hyperpage{165} \item finite moment of order $k$, \hyperpage{184} \item Finite subadditivity, \hyperpage{172} - \item Finite-dimensional distributions, \hyperpage{330} - \item finite-dimensional distributions, \hyperpage{330} - \item finite-rank operator, \hyperpage{305} + \item Finite-dimensional distributions, \hyperpage{332} + \item finite-dimensional distributions, \hyperpage{332} + \item finite-rank operator, \hyperpage{306} \item finitely generated, \hyperpage{164} \item first and second characteristic polynomials, \hyperpage{260} \item First Borel-Cantelli lemma, \hyperpage{187} @@ -907,16 +909,16 @@ \hyperpage{93} \item forward Euler method, \hyperpage{257} \item Forward-time backward-space, \hyperpage{267} - \item Forward-time central-space, \hyperpage{267}, \hyperpage{272} + \item Forward-time central-space, \hyperpage{267}, \hyperpage{273} \item Forward-time forward-space, \hyperpage{267} - \item Fourier coefficients, \hyperpage{311} - \item Fourier series, \hyperpage{82}, \hyperpage{311} + \item Fourier coefficients, \hyperpage{312} + \item Fourier series, \hyperpage{82}, \hyperpage{312} \item Fourier transform, \hyperpage{230}, \hyperpage{236}, - \hyperpage{243}, \hyperpage{311} + \hyperpage{243}, \hyperpage{312} \item Fourier transform operator, \hyperpage{231} - \item Fourier's law, \hyperpage{276} - \item Fredholm alternative, \hyperpage{312} - \item Fredholm operator with kernel $K$, \hyperpage{305} + \item Fourier's law, \hyperpage{277} + \item Fredholm alternative, \hyperpage{313} + \item Fredholm operator with kernel $K$, \hyperpage{306} \item free variables, \hyperpage{13} \item Frenet-Serret formulas, \hyperpage{142} \item Frenet-Serret frame, \hyperpage{142} @@ -927,22 +929,22 @@ \item FTCS, \hyperpage{267} \item FTFS, \hyperpage{267} \item Fubini's theorem, \hyperpage{58}, \hyperpage{174}, - \hyperpage{298} + \hyperpage{299} \item Fubini's theorem for elementary regions, \hyperpage{58} \item full QR decomposition, \hyperpage{266} \item function, \hyperpage{6} \item functionally independent, \hyperpage{133} \item Fundamental lemma of calculus of variations, \hyperpage{239}, - \hyperpage{277} + \hyperpage{278} \item fundamental matrix solution, \hyperpage{126} \item fundamental polygon of the surface, \hyperpage{217} - \item fundamental solution, \hyperpage{244}, \hyperpage{283} + \item fundamental solution, \hyperpage{244}, \hyperpage{284} \item Fundamental theorem of affine geometry, \hyperpage{71} \item Fundamental theorem of algebra, \hyperpage{44}, \hyperpage{113}, \hyperpage{169} \item Fundamental theorem of arithmetic, \hyperpage{9} \item Fundamental theorem of calculus, \hyperpage{32}, - \hyperpage{296} + \hyperpage{297} \item Fundamental theorem of curves, \hyperpage{145} \item Fundamental theorem of Galois theory, \hyperpage{168} \item Fundamental theorem of projective geometry, \hyperpage{67} @@ -954,12 +956,12 @@ \item Galois, \hyperpage{167} \item Galois extension, \hyperpage{167} \item Galois group, \hyperpage{165} - \item Gambler's ruin problem, \hyperpage{316} + \item Gambler's ruin problem, \hyperpage{317} \item Gamma distribution, \hyperpage{178} \item gamma distribution, \hyperpage{178} \item Gamma function, \hyperpage{80} - \item Gau\ss ' law, \hyperpage{276} - \item Gau\ss ' law for magnetism, \hyperpage{276} + \item Gau\ss ' law, \hyperpage{277} + \item Gau\ss ' law for magnetism, \hyperpage{277} \item Gau\ss ' lemma, \hyperpage{46} \item Gau\ss ' theorem, \hyperpage{9, 10}, \hyperpage{13} \item Gau\ss ' Theorema Egregium, \hyperpage{151} @@ -970,8 +972,8 @@ \item Gau\ss \ map, \hyperpage{148} \item Gau\ss ian integers, \hyperpage{46} \item Gaussian elimination, \hyperpage{100} - \item gaussian process, \hyperpage{327} - \item gaussian vector, \hyperpage{327} + \item gaussian process, \hyperpage{329} + \item gaussian vector, \hyperpage{329} \item General Cauchy's integral formula, \hyperpage{114} \item General Cauchy's integral formula for derivatives, \hyperpage{115} @@ -985,7 +987,7 @@ \item Generalized compound probability formula, \hyperpage{173} \item generalized eigenspace, \hyperpage{127} \item generalized eigenvector, \hyperpage{127} - \item generalized heat kernel, \hyperpage{284} + \item generalized heat kernel, \hyperpage{285} \item Generalized Hölder's inequality, \hyperpage{235} \item generalized solution, \hyperpage{244} \item generating set, \hyperpage{15} @@ -1014,13 +1016,13 @@ \item gradient, \hyperpage{55} \item Gradient descent, \hyperpage{263} \item gradient vector field, \hyperpage{60} - \item Gram-Schmidt process, \hyperpage{22}, \hyperpage{311} + \item Gram-Schmidt process, \hyperpage{22}, \hyperpage{312} \item graph, \hyperpage{48}, \hyperpage{54}, \hyperpage{226} \item Graph of a function, \hyperpage{54} \item Great Picard's theorem, \hyperpage{117} \item greatest common divisor, \hyperpage{9}, \hyperpage{45} \item greatest element, \hyperpage{125} - \item Green identities, \hyperpage{277} + \item Green identities, \hyperpage{278} \item Green's formula, \hyperpage{158} \item Green's theorem, \hyperpage{61} \item grid, \hyperpage{267} @@ -1032,25 +1034,25 @@ \indexspace - \item Hahn-Banach theorem, \hyperpage{306} + \item Hahn-Banach theorem, \hyperpage{307} \item half-line, \hyperpage{62} \item half-plane, \hyperpage{62} - \item Hamilton-Jacobi equation, \hyperpage{277} + \item Hamilton-Jacobi equation, \hyperpage{278} \item Hamiltonian, \hyperpage{133} \item Hamiltonian system, \hyperpage{133} \item Handshaking lemma, \hyperpage{48} - \item Hardy-Littlewood maximal function, \hyperpage{296} + \item Hardy-Littlewood maximal function, \hyperpage{297} \item harmonic, \hyperpage{114} \item harmonic ratio, \hyperpage{68} \item Hartman-Grobman theorem, \hyperpage{133}, \hyperpage{221} \item has $t$ correct decimal digits, \hyperpage{88} \item has $u$ significant digits, \hyperpage{88} \item Hausdorff, \hyperpage{211} - \item Heat equation, \hyperpage{139}, \hyperpage{276} + \item Heat equation, \hyperpage{139}, \hyperpage{277} \item heat equation, \hyperpage{139} - \item heat flux, \hyperpage{276} - \item heat kernel, \hyperpage{283} - \item Heaviside step function, \hyperpage{241}, \hyperpage{283} + \item heat flux, \hyperpage{277} + \item heat kernel, \hyperpage{284} + \item Heaviside step function, \hyperpage{241}, \hyperpage{284} \item Heine's theorem, \hyperpage{54} \item Heine-Borel theorem, \hyperpage{213} \item Hermite interpolation problem, \hyperpage{93} @@ -1061,17 +1063,17 @@ \item high-leverage point, \hyperpage{254} \item high-leverage points, \hyperpage{254} \item highest posterior density, \hyperpage{203} - \item Hilbert basis, \hyperpage{310} + \item Hilbert basis, \hyperpage{311} \item Hilbert field, \hyperpage{63} \item Hilbert plane, \hyperpage{63} - \item Hilbert space, \hyperpage{308} + \item Hilbert space, \hyperpage{309} \item Hilbert transform, \hyperpage{246} \item Hilbert's basis theorem, \hyperpage{42} \item Hilbert's Nullstellensatz, \hyperpage{44} - \item Hilbert-Schmidt operator with kernel $K$, \hyperpage{305} + \item Hilbert-Schmidt operator with kernel $K$, \hyperpage{306} \item Hilbert-Schmidt spectral representation theorem, - \hyperpage{311} - \item holding times, \hyperpage{324} + \hyperpage{312} + \item holding times, \hyperpage{326} \item holes, \hyperpage{215} \item holomorphic, \hyperpage{108} \item holomorphic differential equation, \hyperpage{220} @@ -1099,7 +1101,7 @@ \item Hurwitz's theorem, \hyperpage{118} \item hyperbolic, \hyperpage{267}, \hyperpage{272} \item hyperbolic critical point, \hyperpage{133} - \item hyperbolic equations, \hyperpage{275} + \item hyperbolic equations, \hyperpage{276} \item Hyperbolic geometry, \hyperpage{64} \item hyperbolic matrix, \hyperpage{133} \item hyperbolic periodic orbit, \hyperpage{135} @@ -1113,8 +1115,8 @@ \item Hypothesis test, \hyperpage{198} \item hypothesis test, \hyperpage{198} \item Hölder condition, \hyperpage{246} - \item Hölder conjugates, \hyperpage{300} - \item Hölder's inequality, \hyperpage{300} + \item Hölder conjugates, \hyperpage{301} + \item Hölder's inequality, \hyperpage{301} \indexspace @@ -1133,7 +1135,7 @@ \item implicit Euler method, \hyperpage{257} \item implicit form, \hyperpage{121} \item Implicit function theorem, \hyperpage{56} - \item Implicit scheme in finite differences, \hyperpage{284} + \item Implicit scheme in finite differences, \hyperpage{285} \item improper, \hyperpage{36}, \hyperpage{202} \item improper integral, \hyperpage{78} \item in perspective with respect to a line, \hyperpage{67} @@ -1141,15 +1143,15 @@ \item Incidence axioms, \hyperpage{62} \item incidence relation, \hyperpage{62} \item incident, \hyperpage{48} - \item incompressible flow, \hyperpage{275} + \item incompressible flow, \hyperpage{276} \item inconsistent, \hyperpage{13} \item increasing, \hyperpage{27}, \hyperpage{180} \item increment, \hyperpage{180} \item incremental function, \hyperpage{257} - \item independent, \hyperpage{133}, \hyperpage{180} + \item independent, \hyperpage{133}, \hyperpage{180}, \hyperpage{315} \item independent and identically distributed, \hyperpage{180} \item independent events, \hyperpage{173} - \item independent increments, \hyperpage{324} + \item independent increments, \hyperpage{325} \item Independent samples with known variances, \hyperpage{197} \item Independent samples with unknown equal variances, \hyperpage{197} @@ -1160,7 +1162,7 @@ \hyperpage{112}, \hyperpage{114}, \hyperpage{137, 138}, \hyperpage{159} \item indicator function, \hyperpage{8} - \item indistinguishable, \hyperpage{328} + \item indistinguishable, \hyperpage{330} \item induced subgraph, \hyperpage{48} \item Induction axiom, \hyperpage{5} \item Inexact Newton method, \hyperpage{263} @@ -1168,22 +1170,22 @@ \item infinite, \hyperpage{24}, \hyperpage{163} \item infinite order, \hyperpage{240} \item infinite product topology, \hyperpage{208} - \item infinitesimal generator, \hyperpage{325} - \item infinitesimal transition scheme, \hyperpage{325} + \item infinitesimal generator, \hyperpage{327} + \item infinitesimal transition scheme, \hyperpage{327} \item inflection point, \hyperpage{30} \item Information, \hyperpage{255} \item information of an event, \hyperpage{255} \item initial conditions, \hyperpage{121} - \item initial distribution, \hyperpage{316} + \item initial distribution, \hyperpage{317} \item initial point, \hyperpage{214} \item Initial value problem, \hyperpage{121} \item initial value problem, \hyperpage{121} \item initial values, \hyperpage{47} \item injective, \hyperpage{6} - \item inner product, \hyperpage{23}, \hyperpage{81}, \hyperpage{308} + \item inner product, \hyperpage{23}, \hyperpage{81}, \hyperpage{309} \item integrable, \hyperpage{31}, \hyperpage{181, 182} \item integrable function, \hyperpage{58} - \item integrable function over $E$, \hyperpage{295} + \item integrable function over $E$, \hyperpage{296} \item integral, \hyperpage{110}, \hyperpage{157, 158} \item integral curve, \hyperpage{153} \item integral domain, \hyperpage{43} @@ -1191,20 +1193,20 @@ \item Integral in polar coordinates, \hyperpage{59} \item Integral in spherical coordinates, \hyperpage{59} \item integral of $f$ over $\ensuremath {\mathbb {R}}^n$, - \hyperpage{294} - \item integral of $f$ over $E$, \hyperpage{295} + \hyperpage{295} + \item integral of $f$ over $E$, \hyperpage{296} \item integral of $f$ over a measurable set $E\subseteq \ensuremath {\mathbb {R}}^n$, - \hyperpage{294} + \hyperpage{295} \item integral of $f$ over the region $R$, \hyperpage{147} \item integral of $s$ over $\ensuremath {\mathbb {R}}^n$, - \hyperpage{293} - \item integral of $s$ over a measurable set $E$, \hyperpage{293} + \hyperpage{294} + \item integral of $s$ over a measurable set $E$, \hyperpage{294} \item Integral test, \hyperpage{76}, \hyperpage{79} \item Integrating factor, \hyperpage{123} \item integrating factor, \hyperpage{123}, \hyperpage{137} - \item Integration by parts, \hyperpage{32}, \hyperpage{296} + \item Integration by parts, \hyperpage{32}, \hyperpage{297} \item Integration by substitution, \hyperpage{32} - \item inter-arrival times, \hyperpage{324} + \item inter-arrival times, \hyperpage{326} \item Interior, \hyperpage{205} \item interior, \hyperpage{53}, \hyperpage{205} \item interior point, \hyperpage{53}, \hyperpage{156}, @@ -1215,7 +1217,7 @@ \item internally studentized residuals, \hyperpage{253} \item interpolation problem, \hyperpage{92} \item intersection, \hyperpage{5} - \item interval, \hyperpage{289} + \item interval, \hyperpage{290} \item Interval for $\mu $ with $\sigma $ known, \hyperpage{197} \item Intervals for $\mu $ and $\sigma ^2$, \hyperpage{197} \item invariance level, \hyperpage{70} @@ -1237,21 +1239,21 @@ \item inversion, \hyperpage{118} \item Inversion theorem, \hyperpage{232} \item invertible, \hyperpage{6}, \hyperpage{13} - \item Inviscid flow, \hyperpage{275} + \item Inviscid flow, \hyperpage{276} \item involute, \hyperpage{143} \item irreducible, \hyperpage{11}, \hyperpage{43}, \hyperpage{99}, - \hyperpage{326} - \item irreducible chain, \hyperpage{318} - \item irreducible class, \hyperpage{318} + \hyperpage{328} + \item irreducible chain, \hyperpage{319} + \item irreducible class, \hyperpage{319} \item irreducible polynomial, \hyperpage{163} - \item is finer than, \hyperpage{304} + \item is finer than, \hyperpage{305} \item isolated point, \hyperpage{53} \item isolated singular point, \hyperpage{159} \item isolated singularity, \hyperpage{115}, \hyperpage{159} \item isometric, \hyperpage{22} \item isometry, \hyperpage{22}, \hyperpage{73}, \hyperpage{147} \item isomorphic, \hyperpage{16}, \hyperpage{37}, \hyperpage{49}, - \hyperpage{304} + \hyperpage{305} \item isomorphism, \hyperpage{37} \item isomorphism between projective spaces, \hyperpage{66} \item Isoperimetric inequality, \hyperpage{87} @@ -1280,7 +1282,7 @@ \item Jordan form, \hyperpage{20} \item Jordan matrix, \hyperpage{20} \item Jump discontinuity, \hyperpage{27} - \item jump process, \hyperpage{326} + \item jump process, \hyperpage{327} \indexspace @@ -1290,11 +1292,11 @@ \item Klein bottle, \hyperpage{210} \item Kolmogorov, \hyperpage{211} \item Kolmogorov axioms, \hyperpage{172} - \item Kolmogorov extension theorem, \hyperpage{330} + \item Kolmogorov extension theorem, \hyperpage{332} \item Kolmogorov system, \hyperpage{220} - \item Kolmogorov's backward equation, \hyperpage{325} - \item Kolmogorov's continuity theorem, \hyperpage{328} - \item Kolmogorov's forward equation, \hyperpage{325} + \item Kolmogorov's backward equation, \hyperpage{327} + \item Kolmogorov's continuity theorem, \hyperpage{330} + \item Kolmogorov's forward equation, \hyperpage{327} \item Kolmogorov's strong law, \hyperpage{189} \item Kronecker delta, \hyperpage{18} \item Kronecker's lemma, \hyperpage{163} @@ -1311,32 +1313,32 @@ \item Lagrange's interpolation problem, \hyperpage{92} \item Lagrange's theorem, \hyperpage{38} \item Laguerre polynomials, \hyperpage{95} - \item Lamé coefficients, \hyperpage{275} - \item Laplace equation, \hyperpage{139}, \hyperpage{286} + \item Lamé coefficients, \hyperpage{276} + \item Laplace equation, \hyperpage{139}, \hyperpage{287} \item Laplacian, \hyperpage{60} \item lattice of subgroups, \hyperpage{168} \item Laurent series, \hyperpage{116} \item Laurent series theorem, \hyperpage{116} - \item Law of the iterated logarithm, \hyperpage{330} - \item Law of total expectation, \hyperpage{185, 186}, \hyperpage{313} - \item Law of total probability, \hyperpage{173}, \hyperpage{313} + \item Law of the iterated logarithm, \hyperpage{332} + \item Law of total expectation, \hyperpage{185, 186}, \hyperpage{314} + \item Law of total probability, \hyperpage{173}, \hyperpage{314} \item Lax theorem, \hyperpage{260} \item Lax-Friedrichs scheme, \hyperpage{267} - \item Lax-Richtmyer equivalence theorem, \hyperpage{271} + \item Lax-Richtmyer equivalence theorem, \hyperpage{272} \item Lax-Wendroff, \hyperpage{270} \item Lax-Wendroff scheme, \hyperpage{270} \item leading coefficient, \hyperpage{10} - \item Leapfrog scheme, \hyperpage{267}, \hyperpage{272} + \item Leapfrog scheme, \hyperpage{267}, \hyperpage{273} \item least common multiple, \hyperpage{9} \item least element, \hyperpage{8} \item Least-squares method, \hyperpage{249} \item least-squares method, \hyperpage{249} \item leave, \hyperpage{49} - \item Lebesgue differentiation theorem, \hyperpage{297} + \item Lebesgue differentiation theorem, \hyperpage{298} \item Lebesgue integrable, \hyperpage{174} \item Lebesgue integral, \hyperpage{174} - \item Lebesgue measurable, \hyperpage{291, 292} - \item Lebesgue measure, \hyperpage{174}, \hyperpage{291} + \item Lebesgue measurable, \hyperpage{292, 293} + \item Lebesgue measure, \hyperpage{174}, \hyperpage{292} \item left cosets, \hyperpage{38} \item Left side, \hyperpage{119} \item left singular vectors, \hyperpage{264} @@ -1361,7 +1363,7 @@ \item limit, \hyperpage{104}, \hyperpage{187} \item Limit comparison test, \hyperpage{75} \item limit cycle, \hyperpage{135} - \item limit distribution, \hyperpage{326} + \item limit distribution, \hyperpage{328} \item limit inferior, \hyperpage{26}, \hyperpage{187} \item limit of $f$ at infinity, \hyperpage{27} \item limit of the function $f$ at the point $x_0$, \hyperpage{26} @@ -1400,6 +1402,7 @@ \item Little Picard's theorem, \hyperpage{117} \item Liénard system, \hyperpage{227} \item Liénard's theorem, \hyperpage{227} + \item load vector, \hyperpage{275} \item Local behaviour of a holomorphic function, \hyperpage{115} \item local bifurcation, \hyperpage{222} \item local canonical form, \hyperpage{144} @@ -1421,7 +1424,7 @@ \item local truncation error, \hyperpage{261} \item local truncation errors, \hyperpage{257} \item locally, \hyperpage{213} - \item locally bounded, \hyperpage{120}, \hyperpage{302} + \item locally bounded, \hyperpage{120}, \hyperpage{303} \item locally compact, \hyperpage{213} \item locally connected, \hyperpage{214} \item locally integrable, \hyperpage{78} @@ -1462,17 +1465,17 @@ \item Malgrange-Ehrenpreis theorem, \hyperpage{246} \item Mallow's $C_p$ statistic, \hyperpage{255} \item mantissa, \hyperpage{88} - \item Marcinkiewicz interpolation theorem, \hyperpage{304} + \item Marcinkiewicz interpolation theorem, \hyperpage{305} \item marginal pdf, \hyperpage{179} \item marginal pmf, \hyperpage{179} \item Marginal probability density functions, \hyperpage{179} \item marginal probability density functions, \hyperpage{179} \item Marginal probability mass functions, \hyperpage{179} \item marginal probability mass functions, \hyperpage{179} - \item Markov chain, \hyperpage{316} - \item Markov property, \hyperpage{316} + \item Markov chain, \hyperpage{317} + \item Markov property, \hyperpage{317} \item Markov's inequality, \hyperpage{184} - \item material derivative operator, \hyperpage{275} + \item material derivative operator, \hyperpage{276} \item Matrix, \hyperpage{12} \item matrix, \hyperpage{12}, \hyperpage{17} \item matrix exponential, \hyperpage{126} @@ -1485,47 +1488,47 @@ \item Maximum likelihood method, \hyperpage{194} \item Maximum metric, \hyperpage{204} \item Maximum modulus principle, \hyperpage{113} - \item Maximum principle, \hyperpage{285} - \item Maximum principle on unbounded domains, \hyperpage{285} - \item Maxwell-Faraday equation, \hyperpage{276} + \item Maximum principle, \hyperpage{286} + \item Maximum principle on unbounded domains, \hyperpage{286} + \item Maxwell-Faraday equation, \hyperpage{277} \item mean, \hyperpage{177} \item mean curvature, \hyperpage{148} - \item mean function, \hyperpage{327} + \item mean function, \hyperpage{329} \item mean square error, \hyperpage{250} \item mean squared error, \hyperpage{193} \item Mean value property, \hyperpage{114} \item Mean value theorem, \hyperpage{29}, \hyperpage{55} \item Mean value theorem for integrals, \hyperpage{94}, - \hyperpage{296} + \hyperpage{297} \item Mean value theorem for vector-valued functions, \hyperpage{55} \item mean vector, \hyperpage{180}, \hyperpage{195} - \item measurable, \hyperpage{174}, \hyperpage{291, 292} + \item measurable, \hyperpage{174}, \hyperpage{292, 293} \item measurable space, \hyperpage{174} - \item Measure, \hyperpage{289} - \item measure, \hyperpage{173}, \hyperpage{289} + \item Measure, \hyperpage{290} + \item measure, \hyperpage{173}, \hyperpage{290} \item Melnikov's method, \hyperpage{226} \item memoryless, \hyperpage{176, 177} \item meromorphic, \hyperpage{117} \item Mesh, \hyperpage{274} \item mesh, \hyperpage{274} \item mesh-points, \hyperpage{257} - \item Method of characteristics, \hyperpage{277} + \item Method of characteristics, \hyperpage{278} \item Method of moments, \hyperpage{193} \item metric, \hyperpage{204} - \item metric space, \hyperpage{52}, \hyperpage{204}, \hyperpage{298} + \item metric space, \hyperpage{52}, \hyperpage{204}, \hyperpage{299} \item metrizable, \hyperpage{186}, \hyperpage{211} \item Meusnier's theorem, \hyperpage{149} \item minimal, \hyperpage{220} \item minimal element, \hyperpage{8} \item minimal polynomial, \hyperpage{19} \item minimal surface, \hyperpage{148} - \item minimizer, \hyperpage{309} + \item minimizer, \hyperpage{310} \item Minimum modulus principle, \hyperpage{113} - \item Minimum principle, \hyperpage{285} - \item Minimum principle on unbounded domains, \hyperpage{286} + \item Minimum principle, \hyperpage{286} + \item Minimum principle on unbounded domains, \hyperpage{287} \item minimum-variance unbiased estimator, \hyperpage{193} - \item Minkowski inequality, \hyperpage{81}, \hyperpage{301}, - \hyperpage{308} + \item Minkowski inequality, \hyperpage{81}, \hyperpage{302}, + \hyperpage{309} \item Minkowski's integral inequality, \hyperpage{234} \item minor, \hyperpage{15} \item mirror of the reflection, \hyperpage{70} @@ -1543,7 +1546,7 @@ \item Moment-generating function, \hyperpage{185} \item moment-generating function, \hyperpage{185} \item monic, \hyperpage{10} - \item Monotone convergence theorem, \hyperpage{183}, \hyperpage{294} + \item Monotone convergence theorem, \hyperpage{183}, \hyperpage{295} \item monotonic, \hyperpage{25}, \hyperpage{27} \item monotonically decreasing, \hyperpage{25} \item monotonically increasing, \hyperpage{25} @@ -1584,7 +1587,7 @@ \item Nart-Vila theorem, \hyperpage{170} \item Natural cubic spline, \hyperpage{93} - \item Navier-Cauchy equation, \hyperpage{275} + \item Navier-Cauchy equation, \hyperpage{276} \item negation, \hyperpage{7} \item negative, \hyperpage{140} \item negative basis, \hyperpage{141} @@ -1600,7 +1603,7 @@ \item negatively-oriented, \hyperpage{141} \item neighbourhood, \hyperpage{24}, \hyperpage{53}, \hyperpage{206} \item Neumann, \hyperpage{273} - \item Neumann series, \hyperpage{306} + \item Neumann series, \hyperpage{307} \item Neville's algorithm, \hyperpage{92} \item Newton method, \hyperpage{261}, \hyperpage{263} \item Newton's divided differences\\method, \hyperpage{92} @@ -1628,7 +1631,7 @@ \item noncommutative ring, \hyperpage{41} \item nonsingular, \hyperpage{21} \item norm, \hyperpage{23}, \hyperpage{52}, \hyperpage{93}, - \hyperpage{298}, \hyperpage{303} + \hyperpage{299}, \hyperpage{304} \item norm associated with the inner product, \hyperpage{23} \item normal, \hyperpage{37}, \hyperpage{120}, \hyperpage{166}, \hyperpage{189}, \hyperpage{211} @@ -1646,13 +1649,13 @@ \item Normalized power method, \hyperpage{100} \item normalized power method, \hyperpage{100} \item normalizer, \hyperpage{40} - \item normed algebra, \hyperpage{305} - \item normed vector space, \hyperpage{52}, \hyperpage{298} + \item normed algebra, \hyperpage{306} + \item normed vector space, \hyperpage{52}, \hyperpage{299} \item not integrable, \hyperpage{181} \item not orientation-preserving, \hyperpage{216} \item null hypothesis, \hyperpage{198} - \item null recurrent, \hyperpage{322} - \item null set, \hyperpage{173}, \hyperpage{291} + \item null recurrent, \hyperpage{323} + \item null set, \hyperpage{173}, \hyperpage{292} \item number of steps, \hyperpage{267} \item numeric series, \hyperpage{75} \item numeric series of complex numbers, \hyperpage{104} @@ -1666,7 +1669,7 @@ \item objective function, \hyperpage{50} \item observed information, \hyperpage{194} \item odd extension, \hyperpage{83} - \item odd periodic extension, \hyperpage{281} + \item odd periodic extension, \hyperpage{282} \item odds, \hyperpage{256} \item ode, \hyperpage{121} \item Olinde Rodrigues' theorem, \hyperpage{149} @@ -1682,9 +1685,9 @@ \item open cover, \hyperpage{212} \item Open function, \hyperpage{207} \item Open interval, \hyperpage{24} - \item Open mapping theorem, \hyperpage{115}, \hyperpage{307} + \item Open mapping theorem, \hyperpage{115}, \hyperpage{308} \item open sets, \hyperpage{204} - \item operator, \hyperpage{123}, \hyperpage{303} + \item operator, \hyperpage{123}, \hyperpage{304} \item opposite orientations, \hyperpage{141}, \hyperpage{216} \item orbit, \hyperpage{7}, \hyperpage{39}, \hyperpage{129} \item Orbit linear structure, \hyperpage{7} @@ -1714,43 +1717,43 @@ \item oriented vector space, \hyperpage{141}, \hyperpage{216} \item origin, \hyperpage{69} \item orthogonal, \hyperpage{21--23}, \hyperpage{71}, \hyperpage{81}, - \hyperpage{147}, \hyperpage{308} + \hyperpage{147}, \hyperpage{309} \item orthogonal basis, \hyperpage{95} - \item orthogonal complement, \hyperpage{21}, \hyperpage{308} + \item orthogonal complement, \hyperpage{21}, \hyperpage{309} \item orthogonal coordinates, \hyperpage{147} \item orthogonal geometry, \hyperpage{22} \item orthogonal group, \hyperpage{144} \item Orthogonal polynomials, \hyperpage{95} \item orthogonal projection, \hyperpage{22} - \item orthogonal projection on $F$, \hyperpage{309} + \item orthogonal projection on $F$, \hyperpage{310} \item orthogonal reflections, \hyperpage{72} - \item orthogonal system, \hyperpage{310} + \item orthogonal system, \hyperpage{311} \item orthogonal with respect to the weight $\omega (x)$, \hyperpage{95} \item orthonormal, \hyperpage{21}, \hyperpage{81} - \item orthonormal system, \hyperpage{81}, \hyperpage{310} - \item orthonormalization, \hyperpage{311} + \item orthonormal system, \hyperpage{81}, \hyperpage{311} + \item orthonormalization, \hyperpage{312} \item osculating circle, \hyperpage{143} \item Osculating plane, \hyperpage{142} \item osculating sphere, \hyperpage{143} - \item Outer measure, \hyperpage{290} - \item outer measure, \hyperpage{290} + \item Outer measure, \hyperpage{291} + \item outer measure, \hyperpage{291} \item outliars, \hyperpage{254} \item Over-relaxation methods, \hyperpage{99} \indexspace - \item Paley-Wiener-Zygmund theorem, \hyperpage{329} + \item Paley-Wiener-Zygmund theorem, \hyperpage{330} \item Pappus configuration, \hyperpage{67} \item PAQ reduction theorem, \hyperpage{13} \item parabolic, \hyperpage{272} - \item parabolic boundary, \hyperpage{285} - \item parabolic cylinder, \hyperpage{285} + \item parabolic boundary, \hyperpage{286} + \item parabolic cylinder, \hyperpage{286} \item parabolic point, \hyperpage{149} \item parallel, \hyperpage{69}, \hyperpage{151} \item parallel transport, \hyperpage{151} - \item Parallelogram law, \hyperpage{52}, \hyperpage{308} - \item parameter set, \hyperpage{314} + \item Parallelogram law, \hyperpage{52}, \hyperpage{309} + \item parameter set, \hyperpage{315} \item parameter space, \hyperpage{192} \item parametric, \hyperpage{192} \item Parametric bootstrap, \hyperpage{201} @@ -1758,7 +1761,7 @@ \item parametric equations, \hyperpage{69} \item parametrization, \hyperpage{140}, \hyperpage{146} \item parametrized surface, \hyperpage{60} - \item Parseval identity, \hyperpage{311} + \item Parseval identity, \hyperpage{312} \item Parseval's identity, \hyperpage{86} \item partial derivative, \hyperpage{54} \item partial derivative of order $k$, \hyperpage{55} @@ -1789,7 +1792,7 @@ \item percentile confidence interval, \hyperpage{201} \item perfect, \hyperpage{165} \item Perfect fields, \hyperpage{165} - \item period, \hyperpage{318} + \item period, \hyperpage{319} \item Period three theorem, \hyperpage{228} \item Period-doubling bifurcation, \hyperpage{228} \item period-doubling bifurcation, \hyperpage{228} @@ -1802,7 +1805,7 @@ \item permutation matrix, \hyperpage{101} \item Perron-Frobenius theorem, \hyperpage{100} \item perspectivity, \hyperpage{68} - \item pgf, \hyperpage{314} + \item pgf, \hyperpage{315} \item phase portrait, \hyperpage{129} \item phase space, \hyperpage{129} \item Picard iteration process, \hyperpage{124} @@ -1822,7 +1825,7 @@ \item Poincaré compactification, \hyperpage{137} \item Poincaré disk, \hyperpage{137} \item Poincaré index formula, \hyperpage{138} - \item Poincaré inequality, \hyperpage{287} + \item Poincaré inequality, \hyperpage{288} \item Poincaré map, \hyperpage{135} \item Poincaré's method, \hyperpage{225} \item Poincaré-Bendixson theorem, \hyperpage{135} @@ -1830,24 +1833,24 @@ \item Poincaré-Hopf theorem on $S^2$, \hyperpage{138} \item points, \hyperpage{62}, \hyperpage{204} \item points of the quadric, \hyperpage{72} - \item pointwise bounded, \hyperpage{124}, \hyperpage{302} - \item pointwise equicontinuous, \hyperpage{124}, \hyperpage{302} + \item pointwise bounded, \hyperpage{124}, \hyperpage{303} + \item pointwise equicontinuous, \hyperpage{124}, \hyperpage{303} \item Poisson distribution, \hyperpage{176} - \item Poisson equation, \hyperpage{275} + \item Poisson equation, \hyperpage{276} \item Poisson kernel, \hyperpage{230}, \hyperpage{233} - \item Poisson process, \hyperpage{324} + \item Poisson process, \hyperpage{325} \item Poisson summation formula, \hyperpage{237}, \hyperpage{239} \item Polar form, \hyperpage{107} \item polar form, \hyperpage{107} - \item Polarization identity, \hyperpage{308} + \item Polarization identity, \hyperpage{309} \item Pole, \hyperpage{115} \item pole, \hyperpage{115} \item Polya's theorem on $\ensuremath {\mathbb {Z}}$, - \hyperpage{321} + \hyperpage{323} \item Polya's theorem on $\ensuremath {\mathbb {Z}}^2$, - \hyperpage{322} + \hyperpage{323} \item Polya's theorem on $\ensuremath {\mathbb {Z}}^3$, - \hyperpage{322} + \hyperpage{323} \item polyhedron, \hyperpage{50} \item polynomial, \hyperpage{10} \item polynomial ring, \hyperpage{161} @@ -1858,7 +1861,7 @@ \item positive, \hyperpage{140}, \hyperpage{157} \item positive basis, \hyperpage{141} \item positive part, \hyperpage{76} - \item positive recurrent, \hyperpage{322} + \item positive recurrent, \hyperpage{323} \item positive semi-orbit, \hyperpage{129} \item positive-definite, \hyperpage{22} \item positive-semidefinite, \hyperpage{22} @@ -1871,14 +1874,14 @@ \item posterior mean, \hyperpage{202} \item posterior median, \hyperpage{202} \item posterior mode, \hyperpage{202} - \item potential, \hyperpage{60}, \hyperpage{275} + \item potential, \hyperpage{60}, \hyperpage{276} \item power, \hyperpage{198} \item power function, \hyperpage{198} \item Power method, \hyperpage{100} \item power method, \hyperpage{100} \item power series, \hyperpage{77} \item power set, \hyperpage{5} - \item pre-Hilbert space, \hyperpage{308} + \item pre-Hilbert space, \hyperpage{309} \item predicate, \hyperpage{5} \item prediction, \hyperpage{252} \item prediction band, \hyperpage{253} @@ -1899,7 +1902,7 @@ \item principal ideal, \hyperpage{42} \item principal ideal domain, \hyperpage{43} \item principal value, \hyperpage{107} - \item Principle of least action, \hyperpage{277} + \item Principle of least action, \hyperpage{278} \item prior, \hyperpage{201} \item prior distribution, \hyperpage{201} \item probability, \hyperpage{172} @@ -1907,23 +1910,23 @@ \item Probability mass function, \hyperpage{175} \item probability mass function, \hyperpage{175} \item probability space, \hyperpage{172} - \item probability-generating function, \hyperpage{314} + \item probability-generating function, \hyperpage{315} \item product, \hyperpage{10}, \hyperpage{12}, \hyperpage{105} \item product topology, \hyperpage{208} \item products of group subsets, \hyperpage{39} \item projection, \hyperpage{70} - \item Projection theorem, \hyperpage{309} + \item Projection theorem, \hyperpage{310} \item Projections, \hyperpage{70} \item Projective axiom, \hyperpage{65} \item projective frame, \hyperpage{66} \item projective plane, \hyperpage{65} \item projective subvariety, \hyperpage{65} \item projectivity, \hyperpage{68} - \item prolongation, \hyperpage{306} + \item prolongation, \hyperpage{307} \item Propagation of absolute errors, \hyperpage{88} \item Propagation of relative errors, \hyperpage{88} \item proper, \hyperpage{36} - \item proper subspace of $T$, \hyperpage{307} + \item proper subspace of $T$, \hyperpage{308} \item Properties of addition and scalar multiplication of matrices, \hyperpage{12} \item Properties of homotheties, \hyperpage{71} @@ -1934,13 +1937,13 @@ \item Properties of translations, \hyperpage{70} \item Proximity theorem, \hyperpage{228} \item pull-back, \hyperpage{155} - \item pure birth process, \hyperpage{327} - \item pure death process, \hyperpage{327} + \item pure birth process, \hyperpage{328} + \item pure death process, \hyperpage{328} \item purely inseparable, \hyperpage{167} \item purely transcendental, \hyperpage{164} \item Pythagorean, \hyperpage{63} \item Pythagorean plane, \hyperpage{63} - \item Pythagorean theorem, \hyperpage{71}, \hyperpage{308} + \item Pythagorean theorem, \hyperpage{71}, \hyperpage{309} \indexspace @@ -1965,7 +1968,7 @@ \item quotient group, \hyperpage{38} \item quotient map, \hyperpage{209} \item quotient space, \hyperpage{16}, \hyperpage{209}, - \hyperpage{300} + \hyperpage{301} \item quotient space of collapsing a set to a point, \hyperpage{210} \item quotient topology, \hyperpage{209} @@ -1983,20 +1986,20 @@ \item random sample, \hyperpage{192} \item random variable, \hyperpage{174} \item random vector, \hyperpage{178} - \item Random walk, \hyperpage{317} - \item random walk, \hyperpage{317} + \item Random walk, \hyperpage{318} + \item random walk, \hyperpage{318} \item Rank, \hyperpage{13} \item rank, \hyperpage{13}, \hyperpage{15}, \hyperpage{22}, \hyperpage{73} - \item Rankine-Hugoniot equation, \hyperpage{278} + \item Rankine-Hugoniot equation, \hyperpage{279} \item rate, \hyperpage{178} \item rate of convergence, \hyperpage{98} \item Ratio test, \hyperpage{25}, \hyperpage{75} \item ray, \hyperpage{62} \item Rayleigh quotient, \hyperpage{100} \item RC, \hyperpage{63} - \item reachable, \hyperpage{318} - \item real function, \hyperpage{292} + \item reachable, \hyperpage{319} + \item real function, \hyperpage{293} \item real part, \hyperpage{103} \item real random variable, \hyperpage{174} \item realizable, \hyperpage{66} @@ -2007,15 +2010,15 @@ \item rectifiable, \hyperpage{33}, \hyperpage{59}, \hyperpage{140} \item Rectifying plane, \hyperpage{142} \item recurrence relation of order $k$, \hyperpage{47} - \item recurrent, \hyperpage{47}, \hyperpage{320, 321} + \item recurrent, \hyperpage{47}, \hyperpage{321}, \hyperpage{323} \item Reduced row echelon form, \hyperpage{13} \item reduced row echelon form, \hyperpage{13} \item reducible, \hyperpage{99} \item Reducible linear ode of second order, \hyperpage{122} \item reflection, \hyperpage{70}, \hyperpage{119} \item Reflections, \hyperpage{70} - \item reflexive, \hyperpage{306} - \item refractive index, \hyperpage{276} + \item reflexive, \hyperpage{307} + \item refractive index, \hyperpage{277} \item region, \hyperpage{114}, \hyperpage{147}, \hyperpage{203} \item regression axis, \hyperpage{150} \item regression coefficients, \hyperpage{249} @@ -2024,7 +2027,7 @@ \item Regula falsi method, \hyperpage{89} \item regular, \hyperpage{48}, \hyperpage{57}, \hyperpage{130}, \hyperpage{140}, \hyperpage{194, 195}, \hyperpage{211}, - \hyperpage{326} + \hyperpage{327} \item regular distributions, \hyperpage{240} \item regular domain, \hyperpage{147} \item regular region, \hyperpage{159} @@ -2048,6 +2051,7 @@ \item Residues theorem, \hyperpage{116} \item response, \hyperpage{252} \item restrictions, \hyperpage{50} + \item Reuter criterion, \hyperpage{329} \item Riccati differential equation, \hyperpage{122} \item Richardson extrapolation, \hyperpage{94}, \hyperpage{259} \item Riemann conformal representation theorem, \hyperpage{120} @@ -2058,11 +2062,11 @@ \item Riemann-integrable function, \hyperpage{58} \item Riemann-Lebesgue lemma, \hyperpage{84}, \hyperpage{231}, \hyperpage{237} - \item Riesz representation theorem, \hyperpage{310} + \item Riesz representation theorem, \hyperpage{311} \item Riesz theorem, \hyperpage{246} \item Riesz transform, \hyperpage{247} - \item Riesz's theorem, \hyperpage{304} - \item Riesz-Fischer theorem, \hyperpage{311} + \item Riesz's theorem, \hyperpage{305} + \item Riesz-Fischer theorem, \hyperpage{312} \item right cosets, \hyperpage{38} \item Right side, \hyperpage{119} \item right singular vectors, \hyperpage{264} @@ -2112,7 +2116,7 @@ \item SAS criterion, \hyperpage{62, 63} \item scalars, \hyperpage{14} \item scale, \hyperpage{178} - \item Schrödinger equation, \hyperpage{277} + \item Schrödinger equation, \hyperpage{278} \item Schwartz space, \hyperpage{242} \item Schwarz lemma, \hyperpage{119} \item Schwarz theorem, \hyperpage{241} @@ -2129,13 +2133,13 @@ \item Second isomorphism theorem, \hyperpage{17}, \hyperpage{39}, \hyperpage{43} \item Second Sylow theorem, \hyperpage{40} - \item section, \hyperpage{297} + \item section, \hyperpage{298} \item sectorial decomposition, \hyperpage{132} \item segment, \hyperpage{71} \item Segmented regression, \hyperpage{252} - \item self-adjoint, \hyperpage{310} - \item self-conjugate, \hyperpage{302} - \item self-similar, \hyperpage{282} + \item self-adjoint, \hyperpage{311} + \item self-conjugate, \hyperpage{303} + \item self-similar, \hyperpage{283} \item Semi-hyperbolic singular points classification theorem, \hyperpage{222} \item semi-stable, \hyperpage{135} @@ -2144,20 +2148,20 @@ \item semidiscrete Fourier transform, \hyperpage{269} \item Semidiscrete Parseval identity, \hyperpage{269} \item semidynamical system, \hyperpage{129} - \item semilinear, \hyperpage{308} + \item semilinear, \hyperpage{309} \item semilinear isomorphism, \hyperpage{66} - \item Seminorm, \hyperpage{306} - \item seminorm, \hyperpage{93}, \hyperpage{306} + \item Seminorm, \hyperpage{307} + \item seminorm, \hyperpage{93}, \hyperpage{307} \item sensitive dependence on initial conditions, \hyperpage{229} \item Separability theorem, \hyperpage{167} - \item separable, \hyperpage{167}, \hyperpage{299} + \item separable, \hyperpage{167}, \hyperpage{300} \item Separable extension, \hyperpage{167} - \item separate the points, \hyperpage{301} + \item separate the points, \hyperpage{302} \item separated, \hyperpage{206} \item separated by closed neighbourhoods, \hyperpage{206} - \item separating set, \hyperpage{301} - \item Separation of variables, \hyperpage{122}, \hyperpage{281} - \item separation of variables, \hyperpage{281} + \item separating set, \hyperpage{302} + \item Separation of variables, \hyperpage{122}, \hyperpage{282} + \item separation of variables, \hyperpage{282} \item sequence, \hyperpage{53} \item sequence of complex functions, \hyperpage{105} \item sequence of complex numbers, \hyperpage{104} @@ -2191,11 +2195,11 @@ \item simple, \hyperpage{48}, \hyperpage{89}, \hyperpage{115}, \hyperpage{163}, \hyperpage{169}, \hyperpage{198} \item simple curve, \hyperpage{217} - \item simple function, \hyperpage{293} + \item simple function, \hyperpage{294} \item Simple model, \hyperpage{249} \item simple random sample, \hyperpage{192} \item simple random variable, \hyperpage{181} - \item simple random walk, \hyperpage{317} + \item simple random walk, \hyperpage{318} \item Simple ratio, \hyperpage{71} \item simple ratio, \hyperpage{71} \item simple region, \hyperpage{159} @@ -2213,8 +2217,8 @@ \item size, \hyperpage{48}, \hyperpage{198} \item Skorokhod's representation theorem, \hyperpage{189} \item Slutsky's theorem, \hyperpage{189} - \item small change, \hyperpage{277} - \item Sobolev space, \hyperpage{287} + \item small change, \hyperpage{278} + \item Sobolev space, \hyperpage{288} \item solution, \hyperpage{89}, \hyperpage{138} \item solution of a system of equations, \hyperpage{12} \item solution of the ode, \hyperpage{121} @@ -2224,11 +2228,11 @@ \item source, \hyperpage{132} \item space of rapidly decreasing functions, \hyperpage{242} \item special orthogonal group, \hyperpage{144} - \item specific heat capacity, \hyperpage{276} + \item specific heat capacity, \hyperpage{277} \item spectral radius, \hyperpage{97} - \item Spectral theorem, \hyperpage{23}, \hyperpage{311} - \item spectral values, \hyperpage{307} - \item spectrum, \hyperpage{96}, \hyperpage{307} + \item Spectral theorem, \hyperpage{23}, \hyperpage{312} + \item spectral values, \hyperpage{308} + \item spectrum, \hyperpage{96}, \hyperpage{308} \item sphere, \hyperpage{53} \item Spline, \hyperpage{93} \item spline, \hyperpage{93} @@ -2242,13 +2246,13 @@ \item stability region, \hyperpage{260}, \hyperpage{267} \item stabilizer, \hyperpage{39} \item stable, \hyperpage{135}, \hyperpage{260}, \hyperpage{268}, - \hyperpage{272}, \hyperpage{326} + \hyperpage{273}, \hyperpage{327} \item stable degenerated node, \hyperpage{132} \item stable focus, \hyperpage{132} \item stable manifold, \hyperpage{221} \item stable node, \hyperpage{132} \item stable star, \hyperpage{132} - \item standard, \hyperpage{327} + \item standard, \hyperpage{329} \item standard $n$-simplex, \hyperpage{217} \item Standard deviation, \hyperpage{184} \item standard deviation, \hyperpage{177}, \hyperpage{184} @@ -2258,10 +2262,10 @@ \item standard normal distribution, \hyperpage{177} \item standardized form, \hyperpage{178} \item star domain, \hyperpage{114} - \item state space, \hyperpage{314}, \hyperpage{316} - \item states, \hyperpage{316} - \item stationary distribution, \hyperpage{323}, \hyperpage{326} - \item stationary increments, \hyperpage{324} + \item state space, \hyperpage{315}, \hyperpage{317} + \item states, \hyperpage{317} + \item stationary distribution, \hyperpage{324}, \hyperpage{328} + \item stationary increments, \hyperpage{325} \item Statistic, \hyperpage{192} \item statistic, \hyperpage{192} \item statistical model, \hyperpage{192} @@ -2273,17 +2277,18 @@ \item Stereographic projection, \hyperpage{104} \item stereographic projection, \hyperpage{104} \item stiff equations, \hyperpage{260} + \item stiffness matrix, \hyperpage{275} \item Stirling's formula, \hyperpage{80} - \item Stochastic matrix, \hyperpage{316} - \item stochastic matrix, \hyperpage{316} - \item Stochastic process, \hyperpage{314} - \item stochastic process, \hyperpage{314} - \item stochastically equivalent, \hyperpage{328} + \item Stochastic matrix, \hyperpage{317} + \item stochastic matrix, \hyperpage{317} + \item Stochastic process, \hyperpage{315} + \item stochastic process, \hyperpage{315} + \item stochastically equivalent, \hyperpage{330} \item Stokes theorem, \hyperpage{158} \item Stokes' theorem, \hyperpage{61} \item Stolz-Cesàro theorem, \hyperpage{26} - \item Stone-Weierstra\ss \ theorem, \hyperpage{302} - \item stopping time, \hyperpage{319} + \item Stone-Weierstra\ss \ theorem, \hyperpage{303} + \item stopping time, \hyperpage{320} \item strict Lyapunov function, \hyperpage{136} \item strictly concave, \hyperpage{29} \item strictly convex, \hyperpage{29} @@ -2293,23 +2298,23 @@ \item strictly increasing, \hyperpage{25}, \hyperpage{27} \item Strong duality theorem, \hyperpage{51} \item Strong law, \hyperpage{189} - \item Strong Markov property, \hyperpage{319, 320} + \item Strong Markov property, \hyperpage{320, 321} \item strongly consistent estimator, \hyperpage{193} \item Strum's sequence, \hyperpage{91} \item Student's $t$-distribution with $n$ degrees of freedom, \hyperpage{196} \item Sturm's sequence, \hyperpage{91} \item Sturm's theorem, \hyperpage{91} - \item Sturm-Picone comparison theorem, \hyperpage{281} + \item Sturm-Picone comparison theorem, \hyperpage{282} \item sub-multiplicativity, \hyperpage{96} - \item subalgebra, \hyperpage{301} + \item subalgebra, \hyperpage{302} \item subbasis, \hyperpage{205} \item subcover, \hyperpage{212} \item subfield generated, \hyperpage{162} \item Subgroup, \hyperpage{36} \item subgroup, \hyperpage{36} \item subgroup generated, \hyperpage{36} - \item sublinear, \hyperpage{304} + \item sublinear, \hyperpage{305} \item submanifold, \hyperpage{145} \item submanifold with boundary, \hyperpage{156} \item Submersion, \hyperpage{145} @@ -2322,7 +2327,7 @@ \item subsequence, \hyperpage{25} \item subset, \hyperpage{5} \item subspace topology, \hyperpage{207} - \item Substitution principle, \hyperpage{313} + \item Substitution principle, \hyperpage{314} \item success, \hyperpage{176} \item Successive over-relaxation \\method, \hyperpage{99} \item Successive over-relaxation method, \hyperpage{99} @@ -2339,7 +2344,7 @@ \item support points, \hyperpage{92} \item supremum, \hyperpage{24} \item Supremum axiom, \hyperpage{24} - \item supremum norm, \hyperpage{301} + \item supremum norm, \hyperpage{302} \item surface, \hyperpage{57}, \hyperpage{146}, \hyperpage{216} \item surface integral, \hyperpage{60, 61} \item surface intergal, \hyperpage{158} @@ -2395,19 +2400,19 @@ \item The stable manifold theorem, \hyperpage{221} \item theorem, \hyperpage{66} \item theoretical quantiles, \hyperpage{203} - \item thermal conductivity, \hyperpage{276} - \item thermal diffusivity, \hyperpage{276} + \item thermal conductivity, \hyperpage{277} + \item thermal diffusivity, \hyperpage{277} \item thin QR decomposition, \hyperpage{266} \item Third isomorphism theorem, \hyperpage{17}, \hyperpage{39}, \hyperpage{43} \item Third Sylow theorem, \hyperpage{40} \item Tietze extension theorem, \hyperpage{211} - \item time-homogeneous Markov chain, \hyperpage{316} + \item time-homogeneous Markov chain, \hyperpage{317} \item TNB frame, \hyperpage{142} \item tolerance, \hyperpage{254} - \item Tonelli's theorem, \hyperpage{174}, \hyperpage{297} + \item Tonelli's theorem, \hyperpage{174}, \hyperpage{298} \item topological embedding, \hyperpage{213} - \item topological homeomorphism, \hyperpage{304} + \item topological homeomorphism, \hyperpage{305} \item Topological manifold, \hyperpage{215} \item topological manifold, \hyperpage{215} \item topological property, \hyperpage{212} @@ -2425,24 +2430,24 @@ \item torus, \hyperpage{209} \item total mean of squares, \hyperpage{251} \item total order relation, \hyperpage{8} - \item total subset, \hyperpage{299} + \item total subset, \hyperpage{300} \item total sum of squares, \hyperpage{251} \item totally disconnected, \hyperpage{214} \item totally isotropic, \hyperpage{73} \item Tower formula, \hyperpage{164} \item tower of fields, \hyperpage{164} - \item trace, \hyperpage{140}, \hyperpage{287} - \item Trace theorem, \hyperpage{287} - \item Traffic flow equation, \hyperpage{278} + \item trace, \hyperpage{140}, \hyperpage{288} + \item Trace theorem, \hyperpage{288} + \item Traffic flow equation, \hyperpage{279} \item trail, \hyperpage{48} - \item trajectories, \hyperpage{324} + \item trajectories, \hyperpage{325} \item transcendental, \hyperpage{163} \item Transcritical bifurcation, \hyperpage{222} \item transcritical bifurcation, \hyperpage{222} - \item transient, \hyperpage{320, 321} + \item transient, \hyperpage{321}, \hyperpage{323} \item transition functions, \hyperpage{215} - \item transition matrix, \hyperpage{316} - \item transition probabilities, \hyperpage{316}, \hyperpage{323} + \item transition matrix, \hyperpage{317} + \item transition probabilities, \hyperpage{317}, \hyperpage{325} \item transitive, \hyperpage{168} \item translation, \hyperpage{70}, \hyperpage{118} \item Translations, \hyperpage{70} @@ -2457,7 +2462,7 @@ \item triangle, \hyperpage{159} \item Triangular inequality, \hyperpage{24}, \hyperpage{71} \item triangular inequality, \hyperpage{52}, \hyperpage{204}, - \hyperpage{298} + \hyperpage{299} \item triangular system, \hyperpage{96} \item triangularization, \hyperpage{217} \item triple, \hyperpage{89} @@ -2487,18 +2492,18 @@ \item uncorrelated, \hyperpage{184} \item uncountable, \hyperpage{24} \item Unicity of the splitting field, \hyperpage{166} - \item uniform norm, \hyperpage{299}, \hyperpage{301} - \item uniformly bounded, \hyperpage{124}, \hyperpage{302} + \item uniform norm, \hyperpage{300}, \hyperpage{302} + \item uniformly bounded, \hyperpage{124}, \hyperpage{303} \item uniformly continuous, \hyperpage{32}, \hyperpage{54} \item uniformly convergent, \hyperpage{116} - \item uniformly equicontinuous, \hyperpage{124}, \hyperpage{302} + \item uniformly equicontinuous, \hyperpage{124}, \hyperpage{303} \item uniformly most powerful, \hyperpage{199} \item union, \hyperpage{5} \item unique factorization domain, \hyperpage{44} - \item Uniqueness of Dirichlet problem, \hyperpage{286} - \item Uniqueness of the heat equation, \hyperpage{285} + \item Uniqueness of Dirichlet problem, \hyperpage{287} + \item Uniqueness of the heat equation, \hyperpage{286} \item Uniqueness of the heat equation on the unbounded domains, - \hyperpage{286} + \hyperpage{287} \item unit, \hyperpage{42} \item unit normal exterior vector field, \hyperpage{157} \item unit normal vector, \hyperpage{142} @@ -2526,7 +2531,7 @@ \indexspace \item Van der Pol oscillator, \hyperpage{227} - \item vanishes nowhere, \hyperpage{301} + \item vanishes nowhere, \hyperpage{302} \item variable, \hyperpage{10} \item variables, \hyperpage{12} \item Variacions with repetition, \hyperpage{9} @@ -2544,25 +2549,26 @@ \hyperpage{92} \item vector subspace, \hyperpage{15} \item vectors, \hyperpage{14} - \item version, \hyperpage{328} + \item version, \hyperpage{330} \item vertices, \hyperpage{159} \item VIF, \hyperpage{254} - \item Viscid flow, \hyperpage{275} - \item Volterra operator with kernel $K$, \hyperpage{305} - \item volume, \hyperpage{174}, \hyperpage{289} + \item Viscid flow, \hyperpage{276} + \item Volterra operator with kernel $K$, \hyperpage{306} + \item volume, \hyperpage{174}, \hyperpage{290} \item volume element, \hyperpage{155}, \hyperpage{157} \item Volume of revolution, \hyperpage{33} \indexspace \item Wald test, \hyperpage{200} - \item Wald theorem, \hyperpage{313} + \item Wald theorem, \hyperpage{314} \item walk, \hyperpage{48} \item Waring's method, \hyperpage{163} - \item Wave equation, \hyperpage{139}, \hyperpage{275} + \item Wave equation, \hyperpage{139}, \hyperpage{276} \item wave equation, \hyperpage{139} - \item wave function, \hyperpage{277} + \item wave function, \hyperpage{278} \item Weak duality theorem, \hyperpage{50} + \item weak formulation, \hyperpage{274} \item Weak law, \hyperpage{189} \item weakly consistent estimator, \hyperpage{193} \item Weierstra\ss ' theorem, \hyperpage{54}, \hyperpage{113}, @@ -2577,6 +2583,7 @@ \item well-ordered set, \hyperpage{8} \item well-posed, \hyperpage{257} \item well-posed in the Hadamard sense, \hyperpage{257} + \item Wiener process, \hyperpage{329} \item winding number, \hyperpage{112} \item Wintner lemma, \hyperpage{125} \item Wirtinger operators, \hyperpage{110} @@ -2586,7 +2593,7 @@ \indexspace \item Young's convolution inequality, \hyperpage{236} - \item Young's inequality for products, \hyperpage{300} + \item Young's inequality for products, \hyperpage{301} \indexspace diff --git a/main_physics.idx b/main_physics.idx new file mode 100644 index 0000000..f1310ba --- /dev/null +++ b/main_physics.idx @@ -0,0 +1,462 @@ +\indexentry{Charge conservation|hyperpage}{4} +\indexentry{Coulomb's law|hyperpage}{4} +\indexentry{Superposition principle|hyperpage}{4} +\indexentry{electric field|hyperpage}{4} +\indexentry{Superposition principle|hyperpage}{4} +\indexentry{flux|hyperpage}{4} +\indexentry{Gau\ss ' law|hyperpage}{4} +\indexentry{potential difference|hyperpage}{5} +\indexentry{electric potential|hyperpage}{5} +\indexentry{Superposition principle|hyperpage}{5} +\indexentry{electrostatic energy|hyperpage}{5} +\indexentry{Capacitance|hyperpage}{5} +\indexentry{capacitance|hyperpage}{5} +\indexentry{Capacitor|hyperpage}{5} +\indexentry{capacitor|hyperpage}{5} +\indexentry{electric dipole moment|hyperpage}{5} +\indexentry{electric current|hyperpage}{6} +\indexentry{Microscopic Ohm's law|hyperpage}{6} +\indexentry{conductivity|hyperpage}{6} +\indexentry{Macroscopic Ohm's law|hyperpage}{6} +\indexentry{Resistivity|hyperpage}{6} +\indexentry{resistivity|hyperpage}{6} +\indexentry{temperature coefficient of resistivity|hyperpage}{6} +\indexentry{Joule effect|hyperpage}{6} +\indexentry{electromotive force|hyperpage}{6} +\indexentry{emf|hyperpage}{6} +\indexentry{Kirchhoff's laws|hyperpage}{6} +\indexentry{Capacitor discharging|hyperpage}{7} +\indexentry{Capacitor charging|hyperpage}{7} +\indexentry{Lorentz force|hyperpage}{7} +\indexentry{magnetic moment|hyperpage}{7} +\indexentry{Hall effect|hyperpage}{7} +\indexentry{Hall effect|hyperpage}{7} +\indexentry{RC time constant|hyperpage}{7} +\indexentry{vacuum permeability|hyperpage}{8} +\indexentry{Biot-Savart law|hyperpage}{8} +\indexentry{Gau\ss ' law for magnetism|hyperpage}{8} +\indexentry{Ampère's law|hyperpage}{8} +\indexentry{Bohr magneton|hyperpage}{9} +\indexentry{magnetization|hyperpage}{9} +\indexentry{magnetic susceptibility|hyperpage}{9} +\indexentry{ferromagnetic|hyperpage}{9} +\indexentry{paramagnetic|hyperpage}{9} +\indexentry{diamagnetic|hyperpage}{9} +\indexentry{magnetic flux|hyperpage}{9} +\indexentry{Faraday's law|hyperpage}{9} +\indexentry{Lenz's law|hyperpage}{9} +\indexentry{Eddy current|hyperpage}{9} +\indexentry{Eddy currents|hyperpage}{9} +\indexentry{inductor|hyperpage}{10} +\indexentry{mutual inductance|hyperpage}{10} +\indexentry{displacement current|hyperpage}{10} +\indexentry{equation of movement|hyperpage}{11} +\indexentry{average velocity|hyperpage}{11} +\indexentry{instantaneous velocity|hyperpage}{11} +\indexentry{speed|hyperpage}{11} +\indexentry{average acceleration|hyperpage}{11} +\indexentry{instantaneous acceleration|hyperpage}{11} +\indexentry{Uniform linear motion|hyperpage}{11} +\indexentry{Accelerated linear motion|hyperpage}{11} +\indexentry{angular velocity|hyperpage}{11} +\indexentry{angular acceleration|hyperpage}{11} +\indexentry{tangential acceleration|hyperpage}{11} +\indexentry{normal acceleration|hyperpage}{11} +\indexentry{curvature|hyperpage}{11} +\indexentry{radius of curvature|hyperpage}{11} +\indexentry{Curvature|hyperpage}{11} +\indexentry{average curvature|hyperpage}{11} +\indexentry{Arc length|hyperpage}{11} +\indexentry{Projectile motion|hyperpage}{11} +\indexentry{Newton's laws|hyperpage}{12} +\indexentry{Gravity force|hyperpage}{12} +\indexentry{gravity|hyperpage}{12} +\indexentry{Elastic force|hyperpage}{12} +\indexentry{static coefficient of friction|hyperpage}{12} +\indexentry{kinetic coefficient of friction|hyperpage}{12} +\indexentry{Inertial forces|hyperpage}{12} +\indexentry{inertial force|hyperpage}{12} +\indexentry{fictitious force|hyperpage}{12} +\indexentry{Galilean transformation|hyperpage}{12} +\indexentry{Linear momentum of a particle|hyperpage}{13} +\indexentry{linear momentum|hyperpage}{13} +\indexentry{Linear momentum of a system of particles|hyperpage}{13} +\indexentry{linear momentum of the system|hyperpage}{13} +\indexentry{Center of masses|hyperpage}{13} +\indexentry{center of masses|hyperpage}{13} +\indexentry{Angular momentum|hyperpage}{13} +\indexentry{angular momentum|hyperpage}{13} +\indexentry{Torque|hyperpage}{13} +\indexentry{torque|hyperpage}{13} +\indexentry{Mechanical equilibrium|hyperpage}{13} +\indexentry{Work|hyperpage}{13} +\indexentry{work|hyperpage}{13} +\indexentry{Power|hyperpage}{13} +\indexentry{power|hyperpage}{13} +\indexentry{average power|hyperpage}{13} +\indexentry{Kinetic energy|hyperpage}{13} +\indexentry{kinetic energy|hyperpage}{13} +\indexentry{Conservative forces|hyperpage}{13} +\indexentry{conservative|hyperpage}{13} +\indexentry{potential energy|hyperpage}{13} +\indexentry{Potential energy|hyperpage}{13} +\indexentry{Mechanical energy|hyperpage}{14} +\indexentry{Conservation of mechanical energy|hyperpage}{14} +\indexentry{Examples of potential energies|hyperpage}{14} +\indexentry{moment of inertia|hyperpage}{14} +\indexentry{Parallel axis theorem|hyperpage}{14} +\indexentry{inertial frame of reference|hyperpage}{14} +\indexentry{First postulate|hyperpage}{14} +\indexentry{Second postulate|hyperpage}{14} +\indexentry{Lorentz factor|hyperpage}{14} +\indexentry{Lorentz factor|hyperpage}{14} +\indexentry{Time dilation|hyperpage}{14} +\indexentry{proper time|hyperpage}{14} +\indexentry{Length contraction|hyperpage}{15} +\indexentry{proper length|hyperpage}{15} +\indexentry{Lorentz transformations|hyperpage}{15} +\indexentry{Lorentz transformations of velocities|hyperpage}{15} +\indexentry{Matrix form of Lorentz transformations|hyperpage}{15} +\indexentry{Lorentz invariant|hyperpage}{15} +\indexentry{Types of events|hyperpage}{15} +\indexentry{timelike|hyperpage}{15} +\indexentry{lightlike|hyperpage}{15} +\indexentry{spacelike|hyperpage}{15} +\indexentry{spacelike|hyperpage}{15} +\indexentry{Relativistic Doppler effect|hyperpage}{15} +\indexentry{Redshift|hyperpage}{15} +\indexentry{Blueshift|hyperpage}{15} +\indexentry{Relativistic mass|hyperpage}{16} +\indexentry{Relativistic momentum|hyperpage}{16} +\indexentry{Relativistic energy|hyperpage}{16} +\indexentry{Photon energy and momentum|hyperpage}{16} +\indexentry{Lorentz transformations of energy and momentum|hyperpage}{16} +\indexentry{Compton scattering|hyperpage}{16} +\indexentry{fluid|hyperpage}{16} +\indexentry{viscosity|hyperpage}{16} +\indexentry{ideal|hyperpage}{16} +\indexentry{Density|hyperpage}{16} +\indexentry{incompressible|hyperpage}{16} +\indexentry{Pressure|hyperpage}{16} +\indexentry{Hydrostatic pressure|hyperpage}{16} +\indexentry{Pascal's principle|hyperpage}{16} +\indexentry{Archimedes' principle|hyperpage}{16} +\indexentry{buoyancy|hyperpage}{16} +\indexentry{discharge of a fluid|hyperpage}{17} +\indexentry{Continuity equation|hyperpage}{17} +\indexentry{Laminar flow|hyperpage}{17} +\indexentry{Turbulent flow|hyperpage}{17} +\indexentry{Bernolli's principle|hyperpage}{17} +\indexentry{Lift force|hyperpage}{17} +\indexentry{lift coefficient|hyperpage}{17} +\indexentry{Viscosity|hyperpage}{17} +\indexentry{Velocity of a fluid in a channel|hyperpage}{17} +\indexentry{Poiseuille's law|hyperpage}{17} +\indexentry{Resistance in fluids|hyperpage}{17} +\indexentry{Dissipated power|hyperpage}{17} +\indexentry{dissipated power|hyperpage}{17} +\indexentry{Drag forces|hyperpage}{18} +\indexentry{Terminal velocity|hyperpage}{18} +\indexentry{Reynolds number|hyperpage}{18} +\indexentry{Molecular pressure|hyperpage}{20} +\indexentry{Molecular kinetic energy|hyperpage}{20} +\indexentry{Molecular velocities|hyperpage}{20} +\indexentry{mean-square speed|hyperpage}{20} +\indexentry{root-mean-square speed|hyperpage}{20} +\indexentry{Boltzmann distribution|hyperpage}{20} +\indexentry{Boltzmann factor|hyperpage}{20} +\indexentry{Maxwell-Boltzmann distribution|hyperpage}{20} +\indexentry{Distribution of molecular velocities|hyperpage}{20} +\indexentry{Heat capacity|hyperpage}{20} +\indexentry{Specific heat|hyperpage}{20} +\indexentry{Molar heat capacity|hyperpage}{20} +\indexentry{adiabatic index|hyperpage}{21} +\indexentry{Internal energy|hyperpage}{21} +\indexentry{Mayer's relation|hyperpage}{21} +\indexentry{Heat capacities in gases|hyperpage}{21} +\indexentry{Equipartition theorem|hyperpage}{21} +\indexentry{Equipartition theorem on diatomic molecules|hyperpage}{21} +\indexentry{Dulong-Petit law|hyperpage}{21} +\indexentry{Thermal radiation|hyperpage}{21} +\indexentry{Thermal radiation|hyperpage}{21} +\indexentry{Black body|hyperpage}{21} +\indexentry{black body|hyperpage}{21} +\indexentry{Stefan-Boltzmann law|hyperpage}{21} +\indexentry{radiance|hyperpage}{21} +\indexentry{Wien's displacement law|hyperpage}{21} +\indexentry{spectral radiance|hyperpage}{21} +\indexentry{Wien's displacement constant|hyperpage}{21} +\indexentry{Rayleigh-Jeans law|hyperpage}{21} +\indexentry{Rayleigh-Jeans law|hyperpage}{21} +\indexentry{ultraviolet catastrophe|hyperpage}{21} +\indexentry{Planck's law|hyperpage}{22} +\indexentry{Photoelectric effect|hyperpage}{22} +\indexentry{photoelectric effect|hyperpage}{22} +\indexentry{photoelectrons|hyperpage}{22} +\indexentry{threshold frequency|hyperpage}{22} +\indexentry{Stopping potential|hyperpage}{22} +\indexentry{Planck-Einstein relation|hyperpage}{22} +\indexentry{De Broglie relation|hyperpage}{22} +\indexentry{Light|hyperpage}{22} +\indexentry{Compton scattering|hyperpage}{22} +\indexentry{compton scattering|hyperpage}{22} +\indexentry{Bohr's complementary principle|hyperpage}{22} +\indexentry{Wavelenth of a particle|hyperpage}{22} +\indexentry{Heisenberg's uncertainty principle|hyperpage}{22} +\indexentry{Schrödinger equation|hyperpage}{23} +\indexentry{wave function|hyperpage}{23} +\indexentry{Schrödinger equation|hyperpage}{23} +\indexentry{Properties of Schrödinger equation|hyperpage}{23} +\indexentry{Probability density|hyperpage}{23} +\indexentry{Normalization condition|hyperpage}{23} +\indexentry{Expectation value|hyperpage}{23} +\indexentry{Time-independent Schrödinger equation|hyperpage}{23} +\indexentry{time-independent Schrödinger equation|hyperpage}{23} +\indexentry{Particle in a box-1D|hyperpage}{23} +\indexentry{quantum number|hyperpage}{23} +\indexentry{Particle in a box-3D|hyperpage}{23} +\indexentry{degenerate|hyperpage}{23} +\indexentry{Bohr's correspondence principle|hyperpage}{23} +\indexentry{Bohr model|hyperpage}{24} +\indexentry{Radii of Bohr orbits|hyperpage}{24} +\indexentry{first Bohr radius|hyperpage}{24} +\indexentry{Energy levels|hyperpage}{24} +\indexentry{Rydberg-Ritz law|hyperpage}{24} +\indexentry{Rydberg constant|hyperpage}{24} +\indexentry{Quantum numbers in spherical coordinates|hyperpage}{24} +\indexentry{Principal quantum number|hyperpage}{24} +\indexentry{Orbital quantum number|hyperpage}{24} +\indexentry{Magnetic quantum number|hyperpage}{24} +\indexentry{Spin|hyperpage}{24} +\indexentry{intrinsic orbital angular momentum|hyperpage}{24} +\indexentry{spin|hyperpage}{24} +\indexentry{spin quantum number|hyperpage}{24} +\indexentry{Schrödinger equation|hyperpage}{25} +\indexentry{Fermions|hyperpage}{25} +\indexentry{fermion|hyperpage}{25} +\indexentry{Boson|hyperpage}{25} +\indexentry{boson|hyperpage}{25} +\indexentry{Pauli exclusion principle|hyperpage}{25} +\indexentry{Symmetry of the wave function|hyperpage}{25} +\indexentry{Atom|hyperpage}{25} +\indexentry{atomic number|hyperpage}{25} +\indexentry{mass number|hyperpage}{25} +\indexentry{isotopes|hyperpage}{25} +\indexentry{isobars|hyperpage}{25} +\indexentry{isotones|hyperpage}{25} +\indexentry{isomers|hyperpage}{25} +\indexentry{Radii of nucleus|hyperpage}{25} +\indexentry{Nucleus mass|hyperpage}{25} +\indexentry{binding energy|hyperpage}{25} +\indexentry{electron binding energy|hyperpage}{25} +\indexentry{mass defect|hyperpage}{25} +\indexentry{Semi-empirical mass formula|hyperpage}{25} +\indexentry{Q value|hyperpage}{26} +\indexentry{$Q$ value|hyperpage}{26} +\indexentry{$\alpha $-decay|hyperpage}{26} +\indexentry{$\beta $-decay|hyperpage}{26} +\indexentry{$\beta ^-$ decay|hyperpage}{26} +\indexentry{$\beta ^+$ decay|hyperpage}{26} +\indexentry{Electron capture|hyperpage}{26} +\indexentry{$\gamma $-decay|hyperpage}{26} +\indexentry{Radioactive activity|hyperpage}{26} +\indexentry{radioactive activity|hyperpage}{26} +\indexentry{decay constant|hyperpage}{26} +\indexentry{Half-time|hyperpage}{26} +\indexentry{half-time|hyperpage}{26} +\indexentry{Thorium series|hyperpage}{27} +\indexentry{Decay chain|hyperpage}{27} +\indexentry{Secular equilibrium|hyperpage}{27} +\indexentry{Transient equilibrium|hyperpage}{27} +\indexentry{Nuclear reactions|hyperpage}{27} +\indexentry{Exothermic reaction|hyperpage}{27} +\indexentry{Endothermic reaction|hyperpage}{27} +\indexentry{Nuclear fission|hyperpage}{27} +\indexentry{Nuclear fission|hyperpage}{27} +\indexentry{reproduction factor|hyperpage}{27} +\indexentry{Nuclear reactors|hyperpage}{27} +\indexentry{Boiling Water Reactor|hyperpage}{27} +\indexentry{Pressurized Water Reactor|hyperpage}{27} +\indexentry{control roads|hyperpage}{27} +\indexentry{Nuclear fusion|hyperpage}{28} +\indexentry{Nuclear fusion|hyperpage}{28} +\indexentry{Elementary particles|hyperpage}{28} +\indexentry{Antimatter|hyperpage}{28} +\indexentry{antiparticle|hyperpage}{28} +\indexentry{Quark|hyperpage}{28} +\indexentry{quark|hyperpage}{28} +\indexentry{flavors|hyperpage}{28} +\indexentry{up|hyperpage}{28} +\indexentry{down|hyperpage}{28} +\indexentry{charm|hyperpage}{28} +\indexentry{strange|hyperpage}{28} +\indexentry{top|hyperpage}{28} +\indexentry{bottom|hyperpage}{28} +\indexentry{Lepton|hyperpage}{28} +\indexentry{lepton|hyperpage}{28} +\indexentry{flavors|hyperpage}{28} +\indexentry{electron|hyperpage}{28} +\indexentry{electron neutrino|hyperpage}{28} +\indexentry{muon|hyperpage}{28} +\indexentry{muon neutrino|hyperpage}{28} +\indexentry{tau|hyperpage}{28} +\indexentry{tau neutrino|hyperpage}{28} +\indexentry{Lepton number|hyperpage}{28} +\indexentry{Lepton number|hyperpage}{28} +\indexentry{lepton family numbers|hyperpage}{28} +\indexentry{Electron number|hyperpage}{28} +\indexentry{Muon number|hyperpage}{28} +\indexentry{Tau number|hyperpage}{28} +\indexentry{Hadron|hyperpage}{28} +\indexentry{hadron|hyperpage}{28} +\indexentry{Color charge|hyperpage}{28} +\indexentry{color charge|hyperpage}{28} +\indexentry{red|hyperpage}{28} +\indexentry{green|hyperpage}{28} +\indexentry{blue|hyperpage}{28} +\indexentry{antired|hyperpage}{28} +\indexentry{antigreen|hyperpage}{28} +\indexentry{antiblue|hyperpage}{28} +\indexentry{white|hyperpage}{28} +\indexentry{Fundamental interactions|hyperpage}{28} +\indexentry{color confinement|hyperpage}{28} +\indexentry{Grand Unified Theory|hyperpage}{29} +\indexentry{electroweak theory|hyperpage}{29} +\indexentry{Grand Unified Theory|hyperpage}{29} +\indexentry{crystalline|hyperpage}{29} +\indexentry{amorphous|hyperpage}{29} +\indexentry{Classical interpretation of resistivity|hyperpage}{29} +\indexentry{Quantum interpretation of resistivity|hyperpage}{29} +\indexentry{fermi gas|hyperpage}{29} +\indexentry{Fermi energy|hyperpage}{29} +\indexentry{Fermi factor|hyperpage}{29} +\indexentry{Band Theory of Solids|hyperpage}{29} +\indexentry{valence band|hyperpage}{29} +\indexentry{VB|hyperpage}{29} +\indexentry{conduction band|hyperpage}{29} +\indexentry{CB|hyperpage}{29} +\indexentry{Heat|hyperpage}{30} +\indexentry{heat|hyperpage}{30} +\indexentry{conduction|hyperpage}{30} +\indexentry{convection|hyperpage}{30} +\indexentry{radiation|hyperpage}{30} +\indexentry{Conduction|hyperpage}{30} +\indexentry{Thermal conduction|hyperpage}{30} +\indexentry{Fourier's law|hyperpage}{30} +\indexentry{thermal conductivity|hyperpage}{30} +\indexentry{Heat equation|hyperpage}{30} +\indexentry{thermal diffusivity|hyperpage}{30} +\indexentry{Fick's law|hyperpage}{30} +\indexentry{Diffusion equation|hyperpage}{30} +\indexentry{Convection|hyperpage}{30} +\indexentry{Thermal convection|hyperpage}{30} +\indexentry{Newton's law of cooling|hyperpage}{30} +\indexentry{heat transfer coefficient|hyperpage}{30} +\indexentry{Radiation|hyperpage}{31} +\indexentry{Thermal radiation|hyperpage}{31} +\indexentry{Thermodynamic system|hyperpage}{31} +\indexentry{thermodynamic system|hyperpage}{31} +\indexentry{Open system|hyperpage}{31} +\indexentry{Closed system|hyperpage}{31} +\indexentry{Isolated system|hyperpage}{31} +\indexentry{state variables|hyperpage}{31} +\indexentry{Extensive|hyperpage}{31} +\indexentry{Intensive|hyperpage}{31} +\indexentry{specific variables|hyperpage}{31} +\indexentry{mechanical equilibrium|hyperpage}{31} +\indexentry{thermal equilibrium|hyperpage}{31} +\indexentry{chemical equilibrium|hyperpage}{31} +\indexentry{thermodynamical equilibrium|hyperpage}{31} +\indexentry{Stable|hyperpage}{31} +\indexentry{Unstable|hyperpage}{31} +\indexentry{Metastable|hyperpage}{31} +\indexentry{Neutral|hyperpage}{31} +\indexentry{Thermodynamic process|hyperpage}{31} +\indexentry{thermodynamic process|hyperpage}{31} +\indexentry{Quasi-static process|hyperpage}{31} +\indexentry{Reversible process|hyperpage}{31} +\indexentry{Irreversible process|hyperpage}{31} +\indexentry{Non-quasi-static process|hyperpage}{31} +\indexentry{thermal contact|hyperpage}{31} +\indexentry{Zeroth law of thermodynamics|hyperpage}{31} +\indexentry{empirical temperature|hyperpage}{31} +\indexentry{Thermal expansion|hyperpage}{32} +\indexentry{volumetric coefficient of thermal expansion|hyperpage}{32} +\indexentry{Compressibility|hyperpage}{32} +\indexentry{Isothermal compressibility|hyperpage}{32} +\indexentry{Thermal pressure|hyperpage}{32} +\indexentry{thermal pressure coefficient|hyperpage}{32} +\indexentry{Sign convention of work|hyperpage}{32} +\indexentry{First law of thermodynamics in isolated systems|hyperpage}{32} +\indexentry{internal energy|hyperpage}{32} +\indexentry{First law of thermodynamics in closed systems|hyperpage}{32} +\indexentry{heat supplied to the system|hyperpage}{32} +\indexentry{Sign convention of heat|hyperpage}{32} +\indexentry{Latent heat|hyperpage}{32} +\indexentry{Enthalpy|hyperpage}{32} +\indexentry{enthalpy|hyperpage}{32} +\indexentry{Reversible adiabatic equation for an ideal gas|hyperpage}{33} +\indexentry{Second law of thermodynamics|hyperpage}{33} +\indexentry{heat engine|hyperpage}{33} +\indexentry{refrigerator engine|hyperpage}{33} +\indexentry{heat pump|hyperpage}{33} +\indexentry{Carnot cylce|hyperpage}{33} +\indexentry{Carnot cycle|hyperpage}{33} +\indexentry{Carnot's theorem|hyperpage}{33} +\indexentry{Clausius theorem|hyperpage}{34} +\indexentry{Entropy|hyperpage}{34} +\indexentry{entropy|hyperpage}{34} +\indexentry{Second law of thermodynamics in terms of entropy|hyperpage}{34} +\indexentry{Helmholtz free energy|hyperpage}{34} +\indexentry{Gibbs free energy|hyperpage}{34} +\indexentry{Gibbs equation|hyperpage}{34} +\indexentry{Ket|hyperpage}{36} +\indexentry{Hilbert space|hyperpage}{36} +\indexentry{kets|hyperpage}{36} +\indexentry{Bra|hyperpage}{36} +\indexentry{bra|hyperpage}{36} +\indexentry{bracket|hyperpage}{36} +\indexentry{normalized|hyperpage}{36} +\indexentry{orthogonal|hyperpage}{36} +\indexentry{outer product|hyperpage}{36} +\indexentry{linear operator|hyperpage}{36} +\indexentry{adjoint|hyperpage}{36} +\indexentry{daga|hyperpage}{36} +\indexentry{trace|hyperpage}{36} +\indexentry{quantum states|hyperpage}{36} +\indexentry{Hermitian|hyperpage}{37} +\indexentry{self-adjoint|hyperpage}{37} +\indexentry{normal|hyperpage}{37} +\indexentry{degenerate|hyperpage}{37} +\indexentry{spectral decomposition|hyperpage}{37} +\indexentry{Commutator|hyperpage}{37} +\indexentry{commutator|hyperpage}{37} +\indexentry{Anticommutator|hyperpage}{37} +\indexentry{commutator|hyperpage}{37} +\indexentry{compatible|hyperpage}{37} +\indexentry{unitary|hyperpage}{37} +\indexentry{Postulate I|hyperpage}{37} +\indexentry{state vector|hyperpage}{37} +\indexentry{state space|hyperpage}{37} +\indexentry{superposition|hyperpage}{37} +\indexentry{ray|hyperpage}{37} +\indexentry{Postulate II|hyperpage}{37} +\indexentry{observable|hyperpage}{37} +\indexentry{Postulate III (Non-degenerated)|hyperpage}{37} +\indexentry{Postulate III|hyperpage}{37} +\indexentry{Postulate IV|hyperpage}{37} +\indexentry{Postulate V|hyperpage}{38} +\indexentry{Schrödinger equation|hyperpage}{38} +\indexentry{Hamiltonian|hyperpage}{38} +\indexentry{stationary states|hyperpage}{38} +\indexentry{Robertson inequality|hyperpage}{38} +\indexentry{Heisenberg uncertainty principle|hyperpage}{38} +\indexentry{Ehrenfest theorem|hyperpage}{38} +\indexentry{Complete set of commuting observables|hyperpage}{38} +\indexentry{Stern-Gerlach|hyperpage}{38} +\indexentry{gyromagnetic ratio|hyperpage}{38} +\indexentry{Levi-Civita symbol|hyperpage}{38} +\indexentry{Pauli matrices|hyperpage}{38} +\indexentry{Bloch sphere|hyperpage}{38} +\indexentry{qubit|hyperpage}{38} +\indexentry{Bohr magneton|hyperpage}{38} diff --git a/main_physics.ilg b/main_physics.ilg new file mode 100644 index 0000000..0242971 --- /dev/null +++ b/main_physics.ilg @@ -0,0 +1,6 @@ +This is makeindex, version 2.17 [TeX Live 2023] (kpathsea + Thai support). +Scanning input file main_physics.idx....done (462 entries accepted, 0 rejected). +Sorting entries.......done (4646 comparisons). +Generating output file main_physics.ind....done (520 lines written, 0 warnings). +Output written in main_physics.ind. +Transcript written in main_physics.ilg. diff --git a/main_physics.ind b/main_physics.ind new file mode 100644 index 0000000..9221af7 --- /dev/null +++ b/main_physics.ind @@ -0,0 +1,520 @@ +\begin{theindex} + + \item $Q$ value, \hyperpage{26} + \item $\alpha $-decay, \hyperpage{26} + \item $\beta $-decay, \hyperpage{26} + \item $\beta ^+$ decay, \hyperpage{26} + \item $\beta ^-$ decay, \hyperpage{26} + \item $\gamma $-decay, \hyperpage{26} + + \indexspace + + \item Accelerated linear motion, \hyperpage{11} + \item adiabatic index, \hyperpage{21} + \item adjoint, \hyperpage{36} + \item amorphous, \hyperpage{29} + \item Ampère's law, \hyperpage{8} + \item angular acceleration, \hyperpage{11} + \item Angular momentum, \hyperpage{13} + \item angular momentum, \hyperpage{13} + \item angular velocity, \hyperpage{11} + \item antiblue, \hyperpage{28} + \item Anticommutator, \hyperpage{37} + \item antigreen, \hyperpage{28} + \item Antimatter, \hyperpage{28} + \item antiparticle, \hyperpage{28} + \item antired, \hyperpage{28} + \item Arc length, \hyperpage{11} + \item Archimedes' principle, \hyperpage{16} + \item Atom, \hyperpage{25} + \item atomic number, \hyperpage{25} + \item average acceleration, \hyperpage{11} + \item average curvature, \hyperpage{11} + \item average power, \hyperpage{13} + \item average velocity, \hyperpage{11} + + \indexspace + + \item Band Theory of Solids, \hyperpage{29} + \item Bernolli's principle, \hyperpage{17} + \item binding energy, \hyperpage{25} + \item Biot-Savart law, \hyperpage{8} + \item Black body, \hyperpage{21} + \item black body, \hyperpage{21} + \item Bloch sphere, \hyperpage{38} + \item blue, \hyperpage{28} + \item Blueshift, \hyperpage{15} + \item Bohr magneton, \hyperpage{9}, \hyperpage{38} + \item Bohr model, \hyperpage{24} + \item Bohr's complementary principle, \hyperpage{22} + \item Bohr's correspondence principle, \hyperpage{23} + \item Boiling Water Reactor, \hyperpage{27} + \item Boltzmann distribution, \hyperpage{20} + \item Boltzmann factor, \hyperpage{20} + \item Boson, \hyperpage{25} + \item boson, \hyperpage{25} + \item bottom, \hyperpage{28} + \item Bra, \hyperpage{36} + \item bra, \hyperpage{36} + \item bracket, \hyperpage{36} + \item buoyancy, \hyperpage{16} + + \indexspace + + \item Capacitance, \hyperpage{5} + \item capacitance, \hyperpage{5} + \item Capacitor, \hyperpage{5} + \item capacitor, \hyperpage{5} + \item Capacitor charging, \hyperpage{7} + \item Capacitor discharging, \hyperpage{7} + \item Carnot cycle, \hyperpage{33} + \item Carnot cylce, \hyperpage{33} + \item Carnot's theorem, \hyperpage{33} + \item CB, \hyperpage{29} + \item Center of masses, \hyperpage{13} + \item center of masses, \hyperpage{13} + \item Charge conservation, \hyperpage{4} + \item charm, \hyperpage{28} + \item chemical equilibrium, \hyperpage{31} + \item Classical interpretation of resistivity, \hyperpage{29} + \item Clausius theorem, \hyperpage{34} + \item Closed system, \hyperpage{31} + \item Color charge, \hyperpage{28} + \item color charge, \hyperpage{28} + \item color confinement, \hyperpage{28} + \item Commutator, \hyperpage{37} + \item commutator, \hyperpage{37} + \item compatible, \hyperpage{37} + \item Complete set of commuting observables, \hyperpage{38} + \item Compressibility, \hyperpage{32} + \item Compton scattering, \hyperpage{16}, \hyperpage{22} + \item compton scattering, \hyperpage{22} + \item Conduction, \hyperpage{30} + \item conduction, \hyperpage{30} + \item conduction band, \hyperpage{29} + \item conductivity, \hyperpage{6} + \item Conservation of mechanical energy, \hyperpage{14} + \item conservative, \hyperpage{13} + \item Conservative forces, \hyperpage{13} + \item Continuity equation, \hyperpage{17} + \item control roads, \hyperpage{27} + \item Convection, \hyperpage{30} + \item convection, \hyperpage{30} + \item Coulomb's law, \hyperpage{4} + \item crystalline, \hyperpage{29} + \item Curvature, \hyperpage{11} + \item curvature, \hyperpage{11} + + \indexspace + + \item daga, \hyperpage{36} + \item De Broglie relation, \hyperpage{22} + \item Decay chain, \hyperpage{27} + \item decay constant, \hyperpage{26} + \item degenerate, \hyperpage{23}, \hyperpage{37} + \item Density, \hyperpage{16} + \item diamagnetic, \hyperpage{9} + \item Diffusion equation, \hyperpage{30} + \item discharge of a fluid, \hyperpage{17} + \item displacement current, \hyperpage{10} + \item Dissipated power, \hyperpage{17} + \item dissipated power, \hyperpage{17} + \item Distribution of molecular velocities, \hyperpage{20} + \item down, \hyperpage{28} + \item Drag forces, \hyperpage{18} + \item Dulong-Petit law, \hyperpage{21} + + \indexspace + + \item Eddy current, \hyperpage{9} + \item Eddy currents, \hyperpage{9} + \item Ehrenfest theorem, \hyperpage{38} + \item Elastic force, \hyperpage{12} + \item electric current, \hyperpage{6} + \item electric dipole moment, \hyperpage{5} + \item electric field, \hyperpage{4} + \item electric potential, \hyperpage{5} + \item electromotive force, \hyperpage{6} + \item electron, \hyperpage{28} + \item electron binding energy, \hyperpage{25} + \item Electron capture, \hyperpage{26} + \item electron neutrino, \hyperpage{28} + \item Electron number, \hyperpage{28} + \item electrostatic energy, \hyperpage{5} + \item electroweak theory, \hyperpage{29} + \item Elementary particles, \hyperpage{28} + \item emf, \hyperpage{6} + \item empirical temperature, \hyperpage{31} + \item Endothermic reaction, \hyperpage{27} + \item Energy levels, \hyperpage{24} + \item Enthalpy, \hyperpage{32} + \item enthalpy, \hyperpage{32} + \item Entropy, \hyperpage{34} + \item entropy, \hyperpage{34} + \item equation of movement, \hyperpage{11} + \item Equipartition theorem, \hyperpage{21} + \item Equipartition theorem on diatomic molecules, \hyperpage{21} + \item Examples of potential energies, \hyperpage{14} + \item Exothermic reaction, \hyperpage{27} + \item Expectation value, \hyperpage{23} + \item Extensive, \hyperpage{31} + + \indexspace + + \item Faraday's law, \hyperpage{9} + \item Fermi energy, \hyperpage{29} + \item Fermi factor, \hyperpage{29} + \item fermi gas, \hyperpage{29} + \item fermion, \hyperpage{25} + \item Fermions, \hyperpage{25} + \item ferromagnetic, \hyperpage{9} + \item Fick's law, \hyperpage{30} + \item fictitious force, \hyperpage{12} + \item first Bohr radius, \hyperpage{24} + \item First law of thermodynamics in closed systems, \hyperpage{32} + \item First law of thermodynamics in isolated systems, \hyperpage{32} + \item First postulate, \hyperpage{14} + \item flavors, \hyperpage{28} + \item fluid, \hyperpage{16} + \item flux, \hyperpage{4} + \item Fourier's law, \hyperpage{30} + \item Fundamental interactions, \hyperpage{28} + + \indexspace + + \item Galilean transformation, \hyperpage{12} + \item Gau\ss ' law, \hyperpage{4} + \item Gau\ss ' law for magnetism, \hyperpage{8} + \item Gibbs equation, \hyperpage{34} + \item Gibbs free energy, \hyperpage{34} + \item Grand Unified Theory, \hyperpage{29} + \item gravity, \hyperpage{12} + \item Gravity force, \hyperpage{12} + \item green, \hyperpage{28} + \item gyromagnetic ratio, \hyperpage{38} + + \indexspace + + \item Hadron, \hyperpage{28} + \item hadron, \hyperpage{28} + \item Half-time, \hyperpage{26} + \item half-time, \hyperpage{26} + \item Hall effect, \hyperpage{7} + \item Hamiltonian, \hyperpage{38} + \item Heat, \hyperpage{30} + \item heat, \hyperpage{30} + \item Heat capacities in gases, \hyperpage{21} + \item Heat capacity, \hyperpage{20} + \item heat engine, \hyperpage{33} + \item Heat equation, \hyperpage{30} + \item heat pump, \hyperpage{33} + \item heat supplied to the system, \hyperpage{32} + \item heat transfer coefficient, \hyperpage{30} + \item Heisenberg uncertainty principle, \hyperpage{38} + \item Heisenberg's uncertainty principle, \hyperpage{22} + \item Helmholtz free energy, \hyperpage{34} + \item Hermitian, \hyperpage{37} + \item Hilbert space, \hyperpage{36} + \item Hydrostatic pressure, \hyperpage{16} + + \indexspace + + \item ideal, \hyperpage{16} + \item incompressible, \hyperpage{16} + \item inductor, \hyperpage{10} + \item inertial force, \hyperpage{12} + \item Inertial forces, \hyperpage{12} + \item inertial frame of reference, \hyperpage{14} + \item instantaneous acceleration, \hyperpage{11} + \item instantaneous velocity, \hyperpage{11} + \item Intensive, \hyperpage{31} + \item Internal energy, \hyperpage{21} + \item internal energy, \hyperpage{32} + \item intrinsic orbital angular momentum, \hyperpage{24} + \item Irreversible process, \hyperpage{31} + \item isobars, \hyperpage{25} + \item Isolated system, \hyperpage{31} + \item isomers, \hyperpage{25} + \item Isothermal compressibility, \hyperpage{32} + \item isotones, \hyperpage{25} + \item isotopes, \hyperpage{25} + + \indexspace + + \item Joule effect, \hyperpage{6} + + \indexspace + + \item Ket, \hyperpage{36} + \item kets, \hyperpage{36} + \item kinetic coefficient of friction, \hyperpage{12} + \item Kinetic energy, \hyperpage{13} + \item kinetic energy, \hyperpage{13} + \item Kirchhoff's laws, \hyperpage{6} + + \indexspace + + \item Laminar flow, \hyperpage{17} + \item Latent heat, \hyperpage{32} + \item Length contraction, \hyperpage{15} + \item Lenz's law, \hyperpage{9} + \item Lepton, \hyperpage{28} + \item lepton, \hyperpage{28} + \item lepton family numbers, \hyperpage{28} + \item Lepton number, \hyperpage{28} + \item Levi-Civita symbol, \hyperpage{38} + \item lift coefficient, \hyperpage{17} + \item Lift force, \hyperpage{17} + \item Light, \hyperpage{22} + \item lightlike, \hyperpage{15} + \item linear momentum, \hyperpage{13} + \item Linear momentum of a particle, \hyperpage{13} + \item Linear momentum of a system of particles, \hyperpage{13} + \item linear momentum of the system, \hyperpage{13} + \item linear operator, \hyperpage{36} + \item Lorentz factor, \hyperpage{14} + \item Lorentz force, \hyperpage{7} + \item Lorentz invariant, \hyperpage{15} + \item Lorentz transformations, \hyperpage{15} + \item Lorentz transformations of energy and momentum, \hyperpage{16} + \item Lorentz transformations of velocities, \hyperpage{15} + + \indexspace + + \item Macroscopic Ohm's law, \hyperpage{6} + \item magnetic flux, \hyperpage{9} + \item magnetic moment, \hyperpage{7} + \item Magnetic quantum number, \hyperpage{24} + \item magnetic susceptibility, \hyperpage{9} + \item magnetization, \hyperpage{9} + \item mass defect, \hyperpage{25} + \item mass number, \hyperpage{25} + \item Matrix form of Lorentz transformations, \hyperpage{15} + \item Maxwell-Boltzmann distribution, \hyperpage{20} + \item Mayer's relation, \hyperpage{21} + \item mean-square speed, \hyperpage{20} + \item Mechanical energy, \hyperpage{14} + \item Mechanical equilibrium, \hyperpage{13} + \item mechanical equilibrium, \hyperpage{31} + \item Metastable, \hyperpage{31} + \item Microscopic Ohm's law, \hyperpage{6} + \item Molar heat capacity, \hyperpage{20} + \item Molecular kinetic energy, \hyperpage{20} + \item Molecular pressure, \hyperpage{20} + \item Molecular velocities, \hyperpage{20} + \item moment of inertia, \hyperpage{14} + \item muon, \hyperpage{28} + \item muon neutrino, \hyperpage{28} + \item Muon number, \hyperpage{28} + \item mutual inductance, \hyperpage{10} + + \indexspace + + \item Neutral, \hyperpage{31} + \item Newton's law of cooling, \hyperpage{30} + \item Newton's laws, \hyperpage{12} + \item Non-quasi-static process, \hyperpage{31} + \item normal, \hyperpage{37} + \item normal acceleration, \hyperpage{11} + \item Normalization condition, \hyperpage{23} + \item normalized, \hyperpage{36} + \item Nuclear fission, \hyperpage{27} + \item Nuclear fusion, \hyperpage{28} + \item Nuclear reactions, \hyperpage{27} + \item Nuclear reactors, \hyperpage{27} + \item Nucleus mass, \hyperpage{25} + + \indexspace + + \item observable, \hyperpage{37} + \item Open system, \hyperpage{31} + \item Orbital quantum number, \hyperpage{24} + \item orthogonal, \hyperpage{36} + \item outer product, \hyperpage{36} + + \indexspace + + \item Parallel axis theorem, \hyperpage{14} + \item paramagnetic, \hyperpage{9} + \item Particle in a box-1D, \hyperpage{23} + \item Particle in a box-3D, \hyperpage{23} + \item Pascal's principle, \hyperpage{16} + \item Pauli exclusion principle, \hyperpage{25} + \item Pauli matrices, \hyperpage{38} + \item Photoelectric effect, \hyperpage{22} + \item photoelectric effect, \hyperpage{22} + \item photoelectrons, \hyperpage{22} + \item Photon energy and momentum, \hyperpage{16} + \item Planck's law, \hyperpage{22} + \item Planck-Einstein relation, \hyperpage{22} + \item Poiseuille's law, \hyperpage{17} + \item Postulate I, \hyperpage{37} + \item Postulate II, \hyperpage{37} + \item Postulate III, \hyperpage{37} + \item Postulate III (Non-degenerated), \hyperpage{37} + \item Postulate IV, \hyperpage{37} + \item Postulate V, \hyperpage{38} + \item potential difference, \hyperpage{5} + \item Potential energy, \hyperpage{13} + \item potential energy, \hyperpage{13} + \item Power, \hyperpage{13} + \item power, \hyperpage{13} + \item Pressure, \hyperpage{16} + \item Pressurized Water Reactor, \hyperpage{27} + \item Principal quantum number, \hyperpage{24} + \item Probability density, \hyperpage{23} + \item Projectile motion, \hyperpage{11} + \item proper length, \hyperpage{15} + \item proper time, \hyperpage{14} + \item Properties of Schrödinger equation, \hyperpage{23} + + \indexspace + + \item Q value, \hyperpage{26} + \item Quantum interpretation of resistivity, \hyperpage{29} + \item quantum number, \hyperpage{23} + \item Quantum numbers in spherical coordinates, \hyperpage{24} + \item quantum states, \hyperpage{36} + \item Quark, \hyperpage{28} + \item quark, \hyperpage{28} + \item Quasi-static process, \hyperpage{31} + \item qubit, \hyperpage{38} + + \indexspace + + \item radiance, \hyperpage{21} + \item Radiation, \hyperpage{31} + \item radiation, \hyperpage{30} + \item Radii of Bohr orbits, \hyperpage{24} + \item Radii of nucleus, \hyperpage{25} + \item Radioactive activity, \hyperpage{26} + \item radioactive activity, \hyperpage{26} + \item radius of curvature, \hyperpage{11} + \item ray, \hyperpage{37} + \item Rayleigh-Jeans law, \hyperpage{21} + \item RC time constant, \hyperpage{7} + \item red, \hyperpage{28} + \item Redshift, \hyperpage{15} + \item refrigerator engine, \hyperpage{33} + \item Relativistic Doppler effect, \hyperpage{15} + \item Relativistic energy, \hyperpage{16} + \item Relativistic mass, \hyperpage{16} + \item Relativistic momentum, \hyperpage{16} + \item reproduction factor, \hyperpage{27} + \item Resistance in fluids, \hyperpage{17} + \item Resistivity, \hyperpage{6} + \item resistivity, \hyperpage{6} + \item Reversible adiabatic equation for an ideal gas, \hyperpage{33} + \item Reversible process, \hyperpage{31} + \item Reynolds number, \hyperpage{18} + \item Robertson inequality, \hyperpage{38} + \item root-mean-square speed, \hyperpage{20} + \item Rydberg constant, \hyperpage{24} + \item Rydberg-Ritz law, \hyperpage{24} + + \indexspace + + \item Schrödinger equation, \hyperpage{23}, \hyperpage{25}, + \hyperpage{38} + \item Second law of thermodynamics, \hyperpage{33} + \item Second law of thermodynamics in terms of entropy, + \hyperpage{34} + \item Second postulate, \hyperpage{14} + \item Secular equilibrium, \hyperpage{27} + \item self-adjoint, \hyperpage{37} + \item Semi-empirical mass formula, \hyperpage{25} + \item Sign convention of heat, \hyperpage{32} + \item Sign convention of work, \hyperpage{32} + \item spacelike, \hyperpage{15} + \item Specific heat, \hyperpage{20} + \item specific variables, \hyperpage{31} + \item spectral decomposition, \hyperpage{37} + \item spectral radiance, \hyperpage{21} + \item speed, \hyperpage{11} + \item Spin, \hyperpage{24} + \item spin, \hyperpage{24} + \item spin quantum number, \hyperpage{24} + \item Stable, \hyperpage{31} + \item state space, \hyperpage{37} + \item state variables, \hyperpage{31} + \item state vector, \hyperpage{37} + \item static coefficient of friction, \hyperpage{12} + \item stationary states, \hyperpage{38} + \item Stefan-Boltzmann law, \hyperpage{21} + \item Stern-Gerlach, \hyperpage{38} + \item Stopping potential, \hyperpage{22} + \item strange, \hyperpage{28} + \item superposition, \hyperpage{37} + \item Superposition principle, \hyperpage{4, 5} + \item Symmetry of the wave function, \hyperpage{25} + + \indexspace + + \item tangential acceleration, \hyperpage{11} + \item tau, \hyperpage{28} + \item tau neutrino, \hyperpage{28} + \item Tau number, \hyperpage{28} + \item temperature coefficient of resistivity, \hyperpage{6} + \item Terminal velocity, \hyperpage{18} + \item Thermal conduction, \hyperpage{30} + \item thermal conductivity, \hyperpage{30} + \item thermal contact, \hyperpage{31} + \item Thermal convection, \hyperpage{30} + \item thermal diffusivity, \hyperpage{30} + \item thermal equilibrium, \hyperpage{31} + \item Thermal expansion, \hyperpage{32} + \item Thermal pressure, \hyperpage{32} + \item thermal pressure coefficient, \hyperpage{32} + \item Thermal radiation, \hyperpage{21}, \hyperpage{31} + \item Thermodynamic process, \hyperpage{31} + \item thermodynamic process, \hyperpage{31} + \item Thermodynamic system, \hyperpage{31} + \item thermodynamic system, \hyperpage{31} + \item thermodynamical equilibrium, \hyperpage{31} + \item Thorium series, \hyperpage{27} + \item threshold frequency, \hyperpage{22} + \item Time dilation, \hyperpage{14} + \item Time-independent Schrödinger equation, \hyperpage{23} + \item time-independent Schrödinger equation, \hyperpage{23} + \item timelike, \hyperpage{15} + \item top, \hyperpage{28} + \item Torque, \hyperpage{13} + \item torque, \hyperpage{13} + \item trace, \hyperpage{36} + \item Transient equilibrium, \hyperpage{27} + \item Turbulent flow, \hyperpage{17} + \item Types of events, \hyperpage{15} + + \indexspace + + \item ultraviolet catastrophe, \hyperpage{21} + \item Uniform linear motion, \hyperpage{11} + \item unitary, \hyperpage{37} + \item Unstable, \hyperpage{31} + \item up, \hyperpage{28} + + \indexspace + + \item vacuum permeability, \hyperpage{8} + \item valence band, \hyperpage{29} + \item VB, \hyperpage{29} + \item Velocity of a fluid in a channel, \hyperpage{17} + \item Viscosity, \hyperpage{17} + \item viscosity, \hyperpage{16} + \item volumetric coefficient of thermal expansion, \hyperpage{32} + + \indexspace + + \item wave function, \hyperpage{23} + \item Wavelenth of a particle, \hyperpage{22} + \item white, \hyperpage{28} + \item Wien's displacement constant, \hyperpage{21} + \item Wien's displacement law, \hyperpage{21} + \item Work, \hyperpage{13} + \item work, \hyperpage{13} + + \indexspace + + \item Zeroth law of thermodynamics, \hyperpage{31} + +\end{theindex} diff --git a/preamble_formulas.sty b/preamble_formulas.sty index 1bc14b3..f0617a1 100644 --- a/preamble_formulas.sty +++ b/preamble_formulas.sty @@ -196,7 +196,7 @@ % command \grad is by default \nabla in bold face % \renewcommand{\grad}{\operatorname{\vf{grad}}} % gradient \DeclareMathOperator{\rot}{\vf{rot}} % curl -\DeclareMathOperator{\rotp}{\vf{\nabla}\times} % curl for physics +\DeclareMathOperator{\rotp}{\vf{\nabla}\times} % curl for physics \renewcommand{\div}{\operatorname{\vf{div}}} % divergence \DeclareMathOperator{\divp}{\vf{\nabla}\cdot} % divergence for physics % command \laplacian is by default \nabla^2