02_MEF_Vigas_Hermiticos_Funciones_de_Forma.ipynb

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    "# El MEF para vigas. \n",
    "## Funciones de forma hermíticas. \n",
    "\n",
    "Análisis de Estructuras. Grado en Ingeniería Civil. \n",
    "\n",
    "Autor: \n",
    "\n",
    "+ Alejandro E. Martínez-Castro, Esther Puertas García, Rafael Gallego.\n",
    "+ Departamento de Mecánica de Estructuras e Ingeniería Hidráulica. Universidad de Granada.\n",
    "\n",
    "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\"><img alt=\"Licencia de Creative Commons\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-nc-sa/4.0/88x31.png\" /></a><br /><span xmlns:dct=\"http://purl.org/dc/terms/\" property=\"dct:title\">Cuadernos MEF: Funciones de forma hermíticas</span> by <span xmlns:cc=\"http://creativecommons.org/ns#\" property=\"cc:attributionName\">Alejandro Martínez Castro, Esther Puertas García, Rafael Gallego Sevilla</span> is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-nc-sa/4.0/\">Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional License</a>.\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "## Motivación\n",
    "\n",
    "En este cuaderno se presenta un procedimiento sistemático que permite obtener cualquier base de funciones de forma. \n",
    "\n",
    "En primer lugar se obtiene la base hermítica de funciones cúbicas. \n",
    "\n",
    "Finalmente se plantean dos ejercicios, en los que se plantean otras bases de funciones. "
   ]
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    "## Introducción\n",
    "\n",
    "El elemento más utilizado en problemas de flexión de vigas de Bernoulli-Euler (vigas esbeltas) es el cúbico de 2 nodos. Este elemento tiene en cada nodo, 2 grados de libertad (desplazamiento vertical y giro), análogo al elemento-barra (P2D) visto en Cálculo Matricial.  \n",
    "\n",
    "Para este cuaderno se va a emplear el paquete de cálculo simbólico de Python denominado [SymPy](http://www.sympy.org/es/). Dentro del entorno del servidor Jupyter, hay un enlace a SymPy directamente en la pestaña de ayuda \"Help\". \n",
    "\n",
    "Hay varias formas de obtener SymPy\n",
    "\n",
    "- Desde Anaconda, con el comando: \n",
    "      conda install sympy\n",
    "  Este comando puede ejecutarse desde una terminal de Windows (comando CMD), o en Linux / Mac, directamente desde una shell. \n",
    "\n",
    "- Mediante el instalador pip:\n",
    "      pip install sympy "
   ]
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     "text": [
      "IPython console for SymPy 1.0 (Python 2.7.13-32-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://docs.sympy.org/1.0/\n"
     ]
    }
   ],
   "source": [
    "from sympy import init_session\n",
    "init_session()"
   ]
  },
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    "slideshow": {
     "slide_type": "-"
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   "source": [
    "Esto permite que se carguen una serie de variables y funciones por defecto. \n",
    "\n",
    "A continuación se definirá un nuevo símbolo, \"L\", que se empleará para hacer referencia a la longitud del elemento."
   ]
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   "source": [
    "from sympy.matrices import *\n",
    "L = symbols('L')"
   ]
  },
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   "source": [
    "## Determinación de las funciones de forma\n",
    "\n",
    "Consideraremos la notación $h_1(x)$, $h_2(x)$, $h_3(x)$, y $h_4(x)$ para las funciones de forma hermíticas. \n",
    "\n",
    "Estas funciones no forman una base lagrangiana, puesto que las condiciones que se imponen no sólo implican valores puntuales del desplazamiento $w$, sino también del giro $\\theta$. Cada nodo tiene 2 condiciones de contorno. \n",
    "\n",
    "Las funciones de hermite forman base del espacio de polinomios de grado 3. Cada función $h_i(x)$ es combinación lineal de la base de funciones $\\mathcal{B} =  \\lbrace{1, x, x^2, x^3} \\rbrace$.\n",
    "\n",
    "Una función de forma $h_i(x)$ deberá podre escribirse como: \n",
    "\n",
    "\\begin{equation}\n",
    "h_i(x) = \\alpha_1^{(i)} \\cdot 1 + \\alpha_2^{(i)} \\cdot x + \\alpha_3^{(i)} \\cdot x^2 + \\alpha_4^{(i)} \\cdot x^3\n",
    "\\end{equation}\n",
    "\n",
    "siendo $\\alpha_j^(i)$, con $i,j$ variando desde 1 hasta 4, los cuatro coeficientes de la función de forma $i$-ésima. \n",
    "\n",
    "El objetivo es determinar los cuatro coeficientes, teniendo en cuenta las condiciones que se imponen a cada función de forma. \n",
    "\n",
    "- La función $h_1(x)$ debe verificar $h_1(0) = 1$, $h'_1(0) = 0$, $h_1(L) = 0$, $h'_1(L) = 0$.\n",
    "- La función $h_2(x)$ debe verificar $h_2(0) = 0$, $h'_2(0) = 1$, $h_2(L) = 0$, $h'_2(L) = 0$.\n",
    "- La función $h_3(x)$ debe verificar $h_3(0) = 0$, $h'_3(0) = 0$, $h_3(L) = 1$, $h'_3(L) = 0$.\n",
    "- La función $h_4(x)$ debe verificar $h_4(0) = 0$, $h'_4(0) = 0$, $h_4(L) = 0$, $h'_3(L) = 1$.\n"
   ]
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   "source": [
    "Para determinar la primera función de forma, se pueden imponer las cuatro condiciones mediante cuatro ecuaciones. Escribiendo $h_1(x)$ como un producto matriz-vector, se tiene: \n",
    "\n",
    "\\begin{equation}\n",
    "h_1(x) = \\left(1\\, x\\, x^2 \\, x^3 \\right) \\cdot \n",
    "\\begin{pmatrix} \\alpha_1^{(1)} \\\\ \\alpha_2^{(1)} \\\\ \\alpha_3^{(1)} \\\\ \\alpha_4^{(1)} \\end{pmatrix}\n",
    "\\end{equation}"
   ]
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   "source": [
    "Las cuatro condiciones de $h_1(x)$ pueden escribirse en formato matricial evaluando la función y sus derivadas en 0 y L. El sistema se escribirá en formato matricial,\n",
    "\n",
    "\\begin{equation}\n",
    "\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\1 & L & L^{2} & L^{3}\\\\0 & 1 & 2 L & 3 L^{2}\\end{matrix}\\right] \\cdot \n",
    "\\begin{pmatrix} \\alpha_1^{(1)} \\\\ \\alpha_2^{(1)} \\\\ \\alpha_3^{(1)} \\\\ \\alpha_4^{(1)}\\end{pmatrix} =\n",
    "\\begin{pmatrix} 1 \\\\ 0 \\\\ 0 \\\\ 0\\end{pmatrix}\n",
    "\\end{equation}"
   ]
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    "Nótese que en cada fila del producto matriz-vector representa la evaluación de la función de forma $h_1(x)$ o de su derivada $h'_1(x)$ en $x=0$ y $x=L$. La primera fila corresponde a $h_1(0)$. La segunda, a $h'_1(0)$. La tercera, a $h_1(L)$, y la cuarta, a $h'_1(L)$. \n",
    "\n",
    "Resolviendo este sistema, pueden obtenerse los cuatro coeficientes que definen $h_1(x)$. \n",
    "\n",
    "Nótese que este sistema es muy parecido al que se debe plantear para definir los coeficientes de la segunda función de forma. \n",
    "\n",
    "\\begin{equation}\n",
    "\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\1 & L & L^{2} & L^{3}\\\\0 & 1 & 2 L & 3 L^{2}\\end{matrix}\\right] \\cdot \n",
    "\\begin{pmatrix} \\alpha_1^{(2)} \\\\ \\alpha_2^{(2)} \\\\ \\alpha_3^{(2)} \\\\ \\alpha_4^{(2)} \\end{pmatrix} =\n",
    "\\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\\\ 0\\end{pmatrix}\n",
    "\\end{equation}"
   ]
  },
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   "source": [
    "Los cuatro sistemas de ecuaciones (para las cuatro funciones de forma) pueden escribirse a la vez, mediante un producto matricial.\n",
    "\n",
    "\\begin{equation}\n",
    "\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\1 & L & L^{2} & L^{3}\\\\0 & 1 & 2 L & 3 L^{2}\\end{matrix}\\right] \\cdot \n",
    "\\left[\\begin{matrix} \n",
    "\\alpha_1^{(1)} & \\alpha_1^{(2)} & \\alpha_1^{(3)} & \\alpha_1^{(4)}\\\\\n",
    "\\alpha_2^{(1)} & \\alpha_2^{(2)} & \\alpha_2^{(3)} & \\alpha_2^{(4)}\\\\\n",
    "\\alpha_3^{(1)} & \\alpha_3^{(2)} & \\alpha_3^{(3)} & \\alpha_3^{(4)}\\\\\n",
    "\\alpha_4^{(1)} & \\alpha_4^{(2)} & \\alpha_4^{(3)} & \\alpha_4^{(4)}\n",
    "\\end{matrix} \\right]\n",
    " = \\left[\n",
    "\\begin{matrix} \n",
    "1 & 0 & 0 & 0  \\\\\n",
    "0 & 1 & 0 & 0  \\\\\n",
    "0 & 0 & 1 & 0  \\\\\n",
    "0 & 0 & 0 & 1  \n",
    "\\end{matrix} \\right]\n",
    "\\end{equation}\n"
   ]
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   "source": [
    "Denominando \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{M} = \\left[\\begin{matrix}\n",
    "1 & 0 & 0 & 0\\\\\n",
    "0 & 1 & 0 & 0\\\\\n",
    "1 & L & L^{2} & L^{3}\\\\\n",
    "0 & 1 & 2 L & 3 L^{2}\\end{matrix}\\right]\n",
    "\\end{equation}\n",
    "\n",
    "Claramente, la matriz de coeficientes debe ser la inversa de $\\mathbf{M}$, puesto que el producto\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{M} \\cdot \\mathbf{M}^{-1} = \\mathbf{I}\n",
    "\\end{equation}\n",
    "\n",
    "con \n",
    "\n",
    "\\begin{equation}\n",
    "\\mathbf{M}^{-1} = \\left[\\begin{matrix} \n",
    "\\alpha_1^{(1)} & \\alpha_1^{(2)} & \\alpha_1^{(3)} & \\alpha_1^{(4)}\\\\\n",
    "\\alpha_2^{(1)} & \\alpha_2^{(2)} & \\alpha_2^{(3)} & \\alpha_2^{(4)}\\\\\n",
    "\\alpha_3^{(1)} & \\alpha_3^{(2)} & \\alpha_3^{(3)} & \\alpha_3^{(4)}\\\\\n",
    "\\alpha_4^{(1)} & \\alpha_4^{(2)} & \\alpha_4^{(3)} & \\alpha_4^{(4)}\n",
    "\\end{matrix} \\right]\n",
    "\\end{equation}"
   ]
  },
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   "source": [
    "## Determinación de la inversa mediante SymPy\n",
    "\n",
    "A continuación se definirá la matriz $\\mathbf{M}$ simbólica, y se calculará la expresión de su inversa. \n",
    "\n",
    "Esta inversa, por columnas, contiene los coeficientes de las funciones de forma. En primer lugar, se define la base de funciones. "
   ]
  },
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      "text/latex": [
       "$$\\left[\\begin{matrix}1 & x & x^{2} & x^{3}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡       2   3⎤\n",
       "⎣1  x  x   x ⎦"
      ]
     },
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   "source": [
    "funciones = Matrix([[1, x, x**2, x**3]])\n",
    "funciones"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La derivada de esta base se puede calcular también, en formato simbólico. "
   ]
  },
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   "cell_type": "code",
   "execution_count": 4,
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      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 1 & 2 x & 3 x^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡              2⎤\n",
       "⎣0  1  2⋅x  3⋅x ⎦"
      ]
     },
     "execution_count": 4,
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   ],
   "source": [
    "Dfunciones = funciones.diff(x)\n",
    "Dfunciones"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuación se construye la matriz M. En cada fila, contendrá la evaluación de la base y su derivada en $x=0$ y $x=L$. \n",
    "\n",
    "Observe cómo se generan las cuatro filas, y cómo se evalúan las funciones o sus derivadas, según corresponda a cada ecuación. \n",
    "\n",
    "Con la función \n",
    "\n",
    "    Matrix.vstack\n",
    "    \n",
    "Se genera una matriz a partir de las filas. "
   ]
  },
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   "cell_type": "code",
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      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\1 & L & L^{2} & L^{3}\\\\0 & 1 & 2 L & 3 L^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0   0    0  ⎤\n",
       "⎢               ⎥\n",
       "⎢0  1   0    0  ⎥\n",
       "⎢               ⎥\n",
       "⎢       2     3 ⎥\n",
       "⎢1  L  L     L  ⎥\n",
       "⎢               ⎥\n",
       "⎢              2⎥\n",
       "⎣0  1  2⋅L  3⋅L ⎦"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matM = Matrix.vstack(funciones.subs(x,0),Dfunciones.subs(x,0),funciones.subs(x,L),Dfunciones.subs(x,L))\n",
    "matM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "A continuación se evalúa la expresión simbólica de la inversa."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOAAAABkCAMAAACo0swdAAAASFBMVEX///8AAAAAAAAAAAAAAAAA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\n29b8B7DwYmuwDn+9AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\- \\frac{3}{L^{2}} & - \\frac{2}{L} & \\frac{3}{L^{2}} & - \\frac{1}{L}\\\\\\frac{2}{L^{3}} & \\frac{1}{L^{2}} & - \\frac{2}{L^{3}} & \\frac{1}{L^{2}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 1    0    0    0 ⎤\n",
       "⎢                  ⎥\n",
       "⎢ 0    1    0    0 ⎥\n",
       "⎢                  ⎥\n",
       "⎢-3   -2   3    -1 ⎥\n",
       "⎢───  ───  ──   ───⎥\n",
       "⎢  2   L    2    L ⎥\n",
       "⎢ L        L       ⎥\n",
       "⎢                  ⎥\n",
       "⎢2    1    -2   1  ⎥\n",
       "⎢──   ──   ───  ── ⎥\n",
       "⎢ 3    2     3   2 ⎥\n",
       "⎣L    L     L   L  ⎦"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Inv_matM = matM.inv()\n",
    "Inv_matM"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Definición de las funciones de forma\n",
    "\n",
    "La matriz $\\mathbf{M}^{-1}$ contiene, por columnas, los coeficientes que definen las funciones de forma. \n",
    "\n",
    "Para obtener las funciones de forma, basta multiplicar la base de funciones por los coeficientes (columnas) de esta matriz inversa. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "h1 = funciones.dot(Inv_matM.col(0))\n",
    "h2 = funciones.dot(Inv_matM.col(1))\n",
    "h3 = funciones.dot(Inv_matM.col(2))\n",
    "h4 = funciones.dot(Inv_matM.col(3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Función h1(x) = 1 - 3*x**2/L**2 + 2*x**3/L**3\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAAvBAMAAAAybWTAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$1 - \\frac{3 x^{2}}{L^{2}} + \\frac{2 x^{3}}{L^{3}}$$"
      ],
      "text/plain": [
       "       2      3\n",
       "    3⋅x    2⋅x \n",
       "1 - ──── + ────\n",
       "      2      3 \n",
       "     L      L  "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print 'Función h1(x) =', h1\n",
    "h1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Función h2(x) = x - 2*x**2/L + x**3/L**2\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAvBAMAAADX32PpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACZElEQVRIDc1Wv2sTURz/vvy4Xs6YHi10vrZS\ncIhW7R+QweIkLQ5OhUQHt5KAgm4GFbFbwKXRIRGcnOyu2MmtNFCwQxfXgkNitYgWru9+vPe+7917\nrwERvOU+P3MvXL58A3DGtf7RNySc6hWDg2Snlu8giuE+HGCqx26t/EPvwBd4bHCQ7K4WfiIqwzGe\nD+CN5JJg5JvAMrp58IELbsChAi6aHLIIzVUWXmcgezedP+9D5Wsanxhke0xZ6TMk3yttyJ+k0m1Y\nk03G6LdfaTEi30sj3i+/erclm5w9g0s+JypwRzA1v7wQVMJwqHpA5pZu3YXPC0sZhwv1BmlMPHE7\nXMBgBh4FG1jI4qfg+M7Iq2UdqtyBpv9W6zDx/AAIlN4zqtx9eK0oKq1GwmSgypx/50gLvAFcB9jd\n1ppULLO3awjsAFzw/MuQ07+gN7khzBqqkVx4fm+vXe9/ghvaUPH3uWGhpbUSsRSGYXt6bmpfHyJX\nZ6sPLfX/1PKCvzvYP+rHYzbOyfTPt4yZ8qH6vmXMxuqzMSObL+i1sR2V6M9HuqjkdLubh93uompT\nbhmzKM0v/fltY8arMdD3LWMm1yHTj1ezecyUeqafrGbzmJ3Vt6zmuHo/fNnHn4HOH1vW1RwVf+E2\nAKkJnljm1Rwli8cir6DUMq/mKO+Y/ziklmU1035pqDxV0MSyrWaaneyIgoISy7iak3S9obQEjS3L\nao6T4i+IKKYotrSrGUV7FBPEEbRYIvWHwgeCYmSxeKwY7cgBpxhYLBFz6I7MtQRHyGKJlEtff9MX\nHCGLxVPTvaNrvSNOMcDWKdv8ojZt0kFyAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x - \\frac{2 x^{2}}{L} + \\frac{x^{3}}{L^{2}}$$"
      ],
      "text/plain": [
       "       2    3\n",
       "    2⋅x    x \n",
       "x - ──── + ──\n",
       "     L      2\n",
       "           L "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print 'Función h2(x) =', h2\n",
    "h2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Función h3(x) = 3*x**2/L**2 - 2*x**3/L**3\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF4AAAAvBAMAAACYgmA5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBaGUtdLrsMewAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{3 x^{2}}{L^{2}} - \\frac{2 x^{3}}{L^{3}}$$"
      ],
      "text/plain": [
       "   2      3\n",
       "3⋅x    2⋅x \n",
       "──── - ────\n",
       "  2      3 \n",
       " L      L  "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print 'Función h3(x) =', h3\n",
    "h3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Función h4(x) =  -x**2/L + x**3/L**2\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF0AAAAvBAMAAABztds6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{x^{2}}{L} + \\frac{x^{3}}{L^{2}}$$"
      ],
      "text/plain": [
       "   2    3\n",
       "  x    x \n",
       "- ── + ──\n",
       "  L     2\n",
       "       L "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print 'Función h4(x) = ', h4\n",
    "h4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Las funciones de forma tienen, por tanto, las siguientes expresiones: \n",
    "\n",
    "\\begin{align}\n",
    "h_1(x)&  = 1 - \\frac{3 x^{2}}{L^{2}} + \\frac{2 x^{3}}{L^{3}} \\\\\n",
    "h_2(x)& = x - \\frac{2 x^{2}}{L} + \\frac{x^{3}}{L^{2}} \\\\\n",
    "h_3(x)& = \\frac{3 x^{2}}{L^{2}} - \\frac{2 x^{3}}{L^{3}} \\\\\n",
    "h_4(x)& = - \\frac{x^{2}}{L} + \\frac{x^{3}}{L^{2}}\n",
    "\\end{align}\n",
    "\n",
    "Para obtener estas expresiones formateadas se ha usado un comando específico, que devuelve los comandos de LaTeX necesarios para escribir las ecuaciones. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 - \\frac{3 x^{2}}{L^{2}} + \\frac{2 x^{3}}{L^{3}}\n",
      "x - \\frac{2 x^{2}}{L} + \\frac{x^{3}}{L^{2}}\n",
      "\\frac{3 x^{2}}{L^{2}} - \\frac{2 x^{3}}{L^{3}}\n",
      "- \\frac{x^{2}}{L} + \\frac{x^{3}}{L^{2}}\n"
     ]
    }
   ],
   "source": [
    "print latex(h1)\n",
    "print latex(h2)\n",
    "print latex(h3)\n",
    "print latex(h4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dibujo de las funciones de forma hermíticas\n",
    "\n",
    "A continuación se dibujan las funciones de forma. Puesto que éstas dependen de $L$, se ha tomado el intervalo $L=1$ para tener una referencia. \n",
    "\n",
    "La librería SymPy tiene sus propias funciones de dibujo. \n",
    "\n",
    "Para que se muestren dentro del cuaderno, se cargan las opciones de mostrar \"en línea\". "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\ProgramData\\Anaconda2\\lib\\site-packages\\IPython\\core\\magics\\pylab.py:161: UserWarning: pylab import has clobbered these variables: ['Polygon', 'cbrt', 'poly', 'sign', 'flatten', 'conjugate', 'diff', 'tan', 'Circle', 'roots', 'plot', 'eye', 'power', 'floor', 'diag', 'var', 'nan', 'sqrt', 'source', 'add', 'zeros', 'take', 'prod', 'plotting', 'product', 'seterr', 'trace', 'multinomial', 'transpose', 'test', 'ones', 'sinh', 'vectorize', 'cosh', 'sinc', 'mod', 'log', 'cos', 'pi', 'f', 'deprecated', 'tanh', 'det', 'beta', 'sin', 'binomial', 'solve', 'exp', 'reshape', 'trunc', 'invert', 'gamma', 'interactive']\n",
      "`%matplotlib` prevents importing * from pylab and numpy\n",
      "  \"\\n`%matplotlib` prevents importing * from pylab and numpy\"\n"
     ]
    },
    {
     "data": {
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dQf2YSH7z0/O4Or0poSFVng8vmFR5J6goRCTorNh+iD+/t4YlWw/SrlEdHris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\nDgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9a2d0f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%pylab inline\n",
    "from sympy.plotting import plot\n",
    "\n",
    "p1 = plot(h1.subs('L',1),(x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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4RlDNN8ZsEZFpIjKtbrHFQA6QDbwK3F3vLe4F5opIBjAQ+F9XZT2Vnw9fXaKj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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9a81588>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x9654080>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot (h2.subs('L',1),(x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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/1RdfYpzX0UREABWFZ3YfPMyMz7Yye+kOOrRswtCurbnz3L6MSNARhIgEFxVF\nA1u/u4DnP93C/FW7ccCkoZ256axeJHVp7XU0EZGjUlE0gPJKP++uzebvX2wjI6cQPzDltERuPCNR\nz6kWkaCnogig7IISXk/dyawl28kuKKV7XDNuGdebK30JtI6N9jqeiEidqChOsbIKPx9uyOYfqVks\n3pjD8IS29OvYkocvGcK4/vFEakZXEQkxKopTZP2efN5I28VbK3aRd6iMjq2acOu43lw+shs927fw\nOp6ISL2pKE7CpuxCFqzew4L0PcTGRLJ+TwHnDerIFb7unNmnPVGREV5HFBE5aSqKE7Q5u5AF6XtY\nsHoPm3OKMIOUxDi+m9yFCYM70a5FE68jioicUiqK4yguq2DJlv0s3pjLx5tyaRIVweacIlIS4/jd\nRUlMGNyJ+JZNvY4pIhIwKoojlFf6Wbe7gJU7D/LeumyWbs2jrNJPs+hITuvdjguHdObMvu2Jb6Vy\nEJHGIaBFYWYTgKeASOAF59wjR6y36vUXAMXA9c655YHMdKS8Q2Ws3Z3Psm0HSN2Wx4odBzlcXklK\nzzgOFpdx3dgenN0vHl9iW5pG69kPItL4BKwozCwSeAY4D8gClpnZfOfcuhqbTQT6Vv8ZDTxX/d9T\n7nBZJdv3HyIjt4h1uwtYv6eAdXsKyC4oZXTPOJZty2NQl1ZcOao7oxLjGJnYhk6tmgUiiohISAnk\nEUUKkOGc2wJgZnOAi4CaRXER8JJzzgFLzKyNmXV2zu053ptX+h1lFX7KKv0Ul1ZQUFJB/uEy9heV\nkVtUyr7CUrIOHGZHXjHb84rJLSwlOtKo8DsizegT34LTe7dnUJdWDO7aisFd29Ciic7EiYgcKZCf\njF2BnTWWs/jfo4WjbdMVOGZRZOYW0esXC/C7b15LiItlR14xHVs2IbuwFIAIq3q9Y6umjOvXgR7t\nYukeF0vf+Bb0jm+hR4iKiNRRSPwKbWZTgakA0U2a0eLNezEMMzCDXRFGtBn5EUaLCCMqMoKoCKMI\nKAIyPU0fOLm5uXTo0MHrGEFB++Ib2hff0L74Rlpa2hrn3OD6fG0gi2IX0L3Gcrfq1050G5xz04Hp\nAD6fz6Wmpp7apCHK5/OhfVFF++Ib2hff0L74hpmV1PdrA3nr8DKgr5n1NLMY4Cpg/hHbzAemWJUx\nQH5drk+IiEjDCdgRhXOuwszuAN6hanjsDOfcWjO7pXr9NGAhVUNjM6gaHntDoPKIiEj9BPQahXNu\nIVVlUPM3P6a1AAAE20lEQVS1aTX+7oDbT+Q9p06demrChQHti29oX3xD++Ib2hf/ZXp9v9CqPqtD\nSsgFFhEJAvV+xoGmNxURkVoFbVGY2QQz22hmGWZ235HrnXPceeed9OnTh6FDh7J8eYPO/NGgFi1a\nRP/+/enTpw+PPPLI/6x/5ZVXGDp0KEOGDGHs2LGsWrXKg5QN43j74j+WLVtGVFQUb7zxRgOma1h1\n2ReLFy9m2LBhJCUlcfbZZzdwwoZzvH2Rn5/PpEmTSE5OJikpiRdffNGDlIF34403Eh8fz+DBRx8F\nWz1w6M/Vn6urzWxEnd7YORd0f6i6+J0J9AJigFXAoOr1zjnnFixY4CZMmOD8fr/78ssvXUpKigtH\nFRUVrlevXi4zM9OVlpa6oUOHurVr1/7XNp9//rnLy8tzzjm3cOHCRr0v/rPd+PHj3cSJE93rr7/u\nQdLAq8u+OHDggBs4cKDbvn27c8657OxsL6IGXF32xUMPPeTuuece55xzOTk5rm3btq60tNSLuAH1\n8ccfu7S0NJeUlHS01VA1eOjfVJ2GGgN85erwmRysRxRfT//hnCsD/jP9x9fmzZvHlClTMDPGjBnD\nwYMH2bMn/EbWLl26lD59+tCrVy9iYmK46qqrmDdv3n9tM3bsWNq2bQvAmDFjyMrK8iJqwNVlXwA8\n/fTTXHbZZcTHx3uQsmHUZV+8+uqrXHrppSQkJACE7f6oy74wMwoLC3HOUVRURFxcHFFRIXG/8Qk5\n66yziIuLq22Tr6dNcs4tAdqYWefjvnFd2qSh/wCXUzXb7H+WrwX+csQ2/wLOqLH8AeDzOrsX++KI\n7X9ec/tw+lPHn4uuwMdUnVadCVzudW4P98WTVE3MuRhIA6Z4ndvDfdES+Iiq6YGKgAu9zh3A/ZEI\nrDnGunp9boZfpTZiZjYe+AFwhtdZPPQkcK9zzl81i32jFgWMBL4FNAO+NLMlzrlN3sbyxPnASuAc\noDfwnpl96pwr8DZWaAjWojhl03+EgTr9f5rZUOAFYKJzbn8DZWtoddkXPmBOdUm0By4wswrn3NsN\nE7HB1GVfZAH7nXOHgENm9gmQDIRbUdRlX9wAPOKqfo3OMLOtwABgacNEDBr1+twM1msUmv7jG8fd\nF2aWAMwFrg3z3xaPuy+ccz2dc4nOuUTgDeC2MCwJqNu/kXnAGWYWZWaxVM3evL6BczaEuuyLHVQd\nWWFmHYH+wJYGTRkc6vW5GZRHFE7Tf3ytjvvifqAd8Gz1b9IVzjmfV5kDpY77olGoy75wzq03s0XA\nasBP1Xn8Nd6lDow6/lw8CMw0s3SqRvzc65zb51noADGz2cA4oL2ZZQEPANFwcp+boXhntoiINKBg\nPfUkIiJBQkUhIiK1UlGIiEitVBQiIlIrFYWIiNRKRSEiIrVSUYiISK1UFCIiYcrMRlU/d6KpmTU3\ns7VmdvSHVdT2PrrhTkQkfJnZ74GmVE0MmeWc+8MJv4eKQkQkfFXPf7UMKAHGOucqT/Q9dOpJRCS8\ntQNaUPVMjqb1eQMdUYiIhDEzm0/VU0J7Ap2dc3ec6HsE5eyxIiJy8sxsClDunHvVzCKBL8zsHOfc\nhyf0PjqiEBGR2ugahYiI1EpFISIitVJRiIhIrVQUIiJSKxWFiIjUSkUhIiK1UlGIiEitVBQiIlKr\n/w8TZXgSVSEfawAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9cee668>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x9c27518>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot (h3.subs('L',1),(x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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8fEsCw2JDeXDuRk6U6vMxyr18mHyQyHZ+XBQbanWURtECo1qF9oE+TB0Ry/6C\nkzy9QPtjlPs4WFjCt+n53DIoCpuTDw3zU1pgVKsxOCaEJ67oyWdbD/P+9wesjqNUi5ifchCbwI2J\nna2O0miWFBgRCRGRZSKSbv8efJZ2Y0Rkl4hkiMhTtd5PEJF1IrJZRFJEZHDLpVdWevDS7ozoGcZz\n/91BanaR1XGUcqjKqmo+Ssni0h5hRLTzszpOo1l1BvMUsNwYEwcsty+fQUQ8gBnAVUBvYKKI9Lav\n/ivwB2NMAvA7+7JyAzab8PebE2gf6M3UuRs5rv0xyoWtSs8j53gptwzqYnWUJrGqwIwDZttfzwbG\n19FmMJBhjNljjCkH5tm3AzDA6ZHegoBDDsyqnExIgDevTRxA9rFT/Objrdofo1zWf9YfJDTQh1G9\nnHda5HOxqsB0MMacnoQ9B+hQR5tIoPaDD1n29wAeA14UkYPA34CnHRVUOaek6BCevLInX6TmMHvN\nPqvjKNXsco+X8vXOXG5M7IyXR+vsLndYahH5SkRS6/gaV7udqfnzs7F/gj4IPG6MiQIeB2adI8cU\nez9NSl5eXqN/DuW87ru4G6MuCGfO2v1sOXjU6jhKNauPN2ZRVW1a3bMvtTmswBhjRhtj4uv4WgQc\nEZFOAPbvuXV8RDZQ+8h2tr8HMAlYYH/9ETWX086W4y1jTJIxJiksLOx8fyzlRGw24W8390cEHvpg\nE0Ul2h+jXIMxhg+TDzIkJoSYVjKwZV2sOu9aTE2RwP59UR1tkoE4EYkREW9ggn07qOlzudT+eiSQ\n7sCsyokF+3vzt5v6c+R4KU98tFn7Y5RLWLungP0FJUwc3Do790+zqsBMBy4XkXRgtH0ZEYkQkSUA\nxphK4GFgKZAGzDfGbLdvfx/wkohsAf4MTGnh/MqJDOgSzG/H9uKrtFz+9e0eq+Modd4+TD5IW19P\nxsR3tDrKefG0YqfGmAJgVB3vHwLG1lpeAiypo913QKIjM6rWZfKwaNbvLeSFL3cxsEswSdEhVkdS\nqkmOlZTzRWoOEwdF4evlYXWc89I6b01Q6idEhBdu7EfnYD8e/mATBcVlVkdSqkkWbsqmvLK61T77\nUpsWGOUy2vp6MePWgRSWlPPYh5uprtb+GNW6GGP4Ku0I4xMi6B3Rtv4NnJwWGOVS4iOD+P01fVid\nkc+7+nyMamW+Tc/nu4wCLunhGne8WtIHo5QjTRwcRXruCZ77fAc9OrThorjWNcS5cl9vr95LWBsf\nftGvk9V5dWSdAAASy0lEQVRRmoWewSiXIyL8+sqexIUH8si8TRwuOmV1JKXqlZlXzMpdedw+pCs+\nnq27c/80LTDKJfl7e/LP2xIpq6ji4Q82UVFVbXUkpc7p3dX78PawceuQ1t+5f5oWGOWyYsMDmX5D\nPzbsP8r0L3ZaHUepsyo6VcEnG7O4NiGCsDY+VsdpNlpglEu7pn8Ek4dFM+u7vXyx7XD9GyhlgfnJ\nBykpr+Ku4dFWR2lWWmCUy/vt2F4kRLXj1x9vZU9esdVxlDpDZVU1767Zx5CYEPpEBFkdp1lpgVEu\nz9vTxozbBuLlIUydu5FT5VVWR1LqB1+lHSH72CnuGh5jdZRmpwVGuYXIdn68PGEAu3JO8Mry3Too\npnIab3+3j87Bflzeu65psVo3LTDKbVzaI4zfju3FzG/28MH6A1bHUYrU7CLW7ytk8rBoPGxidZxm\npwVGuZV7LophRM8wfr94O5sO6CRlylrvrN6Hv7cHNyW13knFzkULjHIrNpvw8i0JdGjry9S5G3VQ\nTGWZvBNl/HfLIW5K7EyQn5fVcRxCC4xyO+38vZl5eyKFJ8t5ZN4mqnRQTGWBud/vp7yqmknDoq2O\n4jBaYJRbio8M4k/j41mdUcBL/9tldRzlZkorqlidkc9lPcPoFhZodRyH0QKj3NZNSVFMHNyFf67M\nZOn2HKvjKDfyYfJBkvcd5cFLY62O4lBaYJRb+/21vbkoNpS3VmWSqQ9hqhZQVlnFGyszGRQdzKCY\nYKvjOJQlBUZEQkRkmYik27/XeZRF5G0RyRWR1KZsr1R9fDw9+OuN/diXX8KUOSmcKK2wOpJycR9v\nyCLneCmPjIpDxPVuTa7NqjOYp4Dlxpg4YLl9uS7vAmPOY3ul6hXRzo/Xbx3IvoISnpi/RWfCVA5T\nXlnNP1dkMqBLOy6Kdf15iqwqMOOA2fbXs4HxdTUyxqwCCpu6vVINdWH39kwb24v/7TjCP1dmWB1H\nuahPN2WRfeyUW5y9gHUFpoMx5vTQtjlAY8dIaPD2IjJFRFJEJCUvL68JUZW7uGt4NNcNiOSlZbtZ\nsTPX6jjKxVRWVTNjRSb9OgcxwkWmRK6PwwqMiHwlIql1fI2r3c7UDArV5GsS9W1vjHnLGJNkjEkK\nC3OP/6iqaUSEP1/Xl96d2vLIvE3syz9pdSTlQhZtPsSBwhJ+OdI9zl7AgQXGGDPaGBNfx9ci4IiI\ndAKwf2/sn4vnu71SdfLz9mDm7Yl42oQp76VQrJ3+qhlUVRtmrMigV6e2jO4VbnWcFmPVJbLFwCT7\n60nAohbeXqmzigrx5/VbB9I+wIcnPtJOf3X+Ptt6iD35J3l0VKzbnL2AdQVmOnC5iKQDo+3LiEiE\niCw53UhE/gOsBXqKSJaI3HOu7ZVqLsNjQxnduwNLtx/h5eXpVsdRrVh1teG1rzPo2aENV/TuaHWc\nFuVpxU6NMQXAqDrePwSMrbU8sTHbK9Wc7h4ezc7Dx3l1eToXdGzD2L6drI6kWqEvUnPIyC3mtYkD\nsLngkPznok/yK3UWIsKfrotnYJd2PDF/C9sPFVkdSbUyNWcv6XQPC3DLP1C0wCh1Dj6eHsy8I5Eg\nPy+mzNlAvg7vrxphWdoRduac4Jcj41xyQrH6aIFRqh7hbXx5685E8ovLmPr+Rsorq62OpFoBYwyv\nLk8nJjSAq/u539kLaIFRqkH6dW7HX2/sR3llFX/8bDs1j18pdXb/234Ef28PfjkyFk8P9/xV654/\ntVJNMC4hkuFxoby/7gCzvttrdRzlxMorq/nLF2kUnarg2v4RVsexjCV3kSnVWj1xeU/25p/k+SVp\nRLcPYHTvxo5ypNzBe+v2s6+ghHfvGuS2Zy+gZzBKNYrNJrx0UwJ9I4N4ZN4mdhw6bnUk5WSOlZTz\n6vJ0LukRxoie7vPUfl20wCjVSH7eHvz7ziSC/Ly4d3YyuSdKrY6knMgry9M5UVrBtLG9rI5iOS0w\nSjVBeFtf/nVnEkdLKnjgvQ2cKq+0OpJyAnvyinlv7X4mDO5Cz45trI5jOS0wSjVRfGQQr0xIoKLK\n8CudqEwB07/YiY+njcdH97A6ilPQAqPUebiiT0fGJUTwRWoOf16SZnUcZaG1mQX8b8cRpl4WS1gb\nH6vjOAW9i0yp83TPRTFkHT3Fv7/bS1SIP5OGRVsdSbWw6mrD80t2ENnOj3suirE6jtPQAqPUeRIR\nnrm6N1lHT/GH/24nop0fl+vty27l003ZpGYf55UJCfh6eVgdx2noJTKlmoGHTXh1YgLxkUE8Pm8T\nWw4eszqSaiEl5ZW8uHQX/aPaufVDlXXRAqNUM/H39mTWpEEMimnPXe8msyev2OpIqgXM+m4vOcdL\neeYXvdxqMrGG0AKjVDMKa+PD767pjQB3vr2e3OP6jIwr25t/kpkrM5lycTeSokOsjuN0tMAo1cxi\nQgN4e/IgCk+WM+mdZI6XVlgdSTmAMYanF2zFZhPuvVg79uuiBUYpB+gf1Y43bk8k/cgJ7p+zgbLK\nKqsjqWY2P+Ug6/YUMm1sL8Lb+lodxylZUmBEJERElolIuv178FnavS0iuSKS+pP3XxSRnSKyVUQ+\nFZF2LZNcqYa7tEcYL97Uj7V7CvjT52lU6YOYLiP3RCnPf57GkJgQbhkUZXUcp2XVGcxTwHJjTByw\n3L5cl3eBMXW8vwyIN8b0A3YDTzsipFLn67oBnXn+unj+8/0Bpn26TeeRcRF/WLyD0spq/nJ9X+3Y\nPwerCsw4YLb99WxgfF2NjDGrgMI63v+fMeb04E/rgM6OCKlUc7htSFceHNGdeckH+dPnaVpkWrll\nO47w+bbDPDoqjm5hgVbHcWpWPWjZwRhz2P46Bzifp9LuBj4820oRmQJMAejSpct57EappvvV5T04\nUVrJrO/20sbXk8d0rKpW6URpBc8sTOWCjm2Yckk3q+M4PYcVGBH5CuhYx6pptReMMUZEmvQnnYhM\nAyqBuWdrY4x5C3gLICkpSf90VJYQEX53dW+Kyyp5+at0An08ufdi/QXV2vz1y10cOVHKzDsS8XLj\nicQaymEFxhgz+mzrROSIiHQyxhwWkU5AbmM/X0QmA1cDo4xec1CtgM0mTL++LyfLKvnT52kE+ngw\nYXBXq2OpBlqTkc+G/Ue5a1gMCVF6X1FDWFWCFwOT7K8nAYsas7GIjAGeBK41xpQ0czalHMbTw8bL\nExK4oncH3l93gAUbs6yOpBrg6MlyfjV/C6UVlfy/K/TyZkNZVWCmA5eLSDow2r6MiESIyJLTjUTk\nP8BaoKeIZInIPfZVrwNtgGUisllEZrZsfKWazsfTg1cmDCDI34snPtrCp5u0yDgzYwy//ngLhSfL\neXXiQPx9dIzghrLkSBljCoBRdbx/CBhba3niWbaPdVw6pRyvZtrlQdz9bjJPzN+CTYRxCZFWx1J1\nmL1mH1+l5fK7q3sTHxlkdZxWRXuplLKIn7cHsyYnMTgmhMc/3MziLYesjqR+YvuhIv68ZCejLgjn\nruHRVsdpdbTAKGUhf29P3p48iKToEP7+v10s3JRtdSRlV1JeyS//s4ngAC9evKm/PlDZBFpglLKY\nv7cn70weRI8ObXh8/mbmrT9gdSQFPLtoO3vzT/KPWxIICfC2Ok6rpAVGKScQ4OPJqxMHcGmPMJ5a\nsI23v9trdSS3tmhzNh9tyOLhy2IZ1j3U6jitlhYYpZyEr5cHb92RxFXxHfnjZzuYsSLD6khuKSP3\nBHPW7iepazCPjoqzOk6rpgVGKSfi7WnjtYkDuG5AJC8u3cWMFRk6dlkLyi8uY/I7yRwsPMnrtw7E\nU5/WPy96Q7dSTsbTw8ZLN/Wna4gfLy7dRWZeMS/c0E+HJnGw0ooq7puTQn5xGR9OuZCOQTrHy/nS\nAqOUE7LZhEdH98Bms/H3ZbvJLy7njdsGEqAP+TlEdbXhiY+2sPngMd64bSD9dSiYZqF/EinlpESE\nR0bFMf36vqzOyGfCW+vIO1FmdSyX9NKyXXy+9TBPjbmAMfGdrI7jMrTAKOXkJgzuwlt3JJKee4Ib\n3ljNntxiqyO5lPkpB5mxIpOJg6N0CP5mpgVGqVZgVK8O/Oe+oXQPC2T8P1fzbXqe1ZFcwpqMfH67\nYBsXx4Xyx3Hx+jBlM9MCo1QrMaBLMH8cF0+nID8mv5PMO6v36h1m5yHtcBG/mr+FmNAAZtw2UG+i\ncAA9okq1IlEh/nwydRijLgjnD//dwVOfbKOsssrqWK3OjkPHue3f64kO9eftyYNo6+tldSSXpAVG\nqVYm0MeTmbcn8sjIWD5MOcht//peO/8bYVtWERP/tQ5fTxvTr+9HVIi/1ZFclhYYpVohm0341RU9\nef3WAaQeKuLXH21hw/6jVsdyepsOHOXWf68j0MeTD++/kOjQAKsjuTQtMEq1Ylf3i2DBg8PJOnqK\nm99cy8xvMqmu1n6ZumzYX8gds9YT7O/N/Acu1DOXFqAFRqlWrndEWxY8NIwr+3Rg+hc7mfxuMvnF\nesmstu/3FHDnrPWEtfHhw/uHEtnOz+pIbkELjFIuoK2vFzNuHcifxsezbk8BY1/5lrUZBVbHcgoL\nN2Xz16W7iAz2Y96UoXQK0uLSUiwpMCISIiLLRCTd/j34LO3eFpFcEUk9y/onRMSIiI6nrdyeiHD7\n0K4snDqczsF+PPjBBp5dlMrJskqro1misqqa5z7bwWMfbsbTJsybMpQObXV8sZZk1RnMU8ByY0wc\nsNy+XJd3gTF1rRCRKOAKQGdnUqqW3hFtef/eIYxPiGT22v2MeWUVazLyrY7VogqKy7hj1npmfbeX\nycOief/eIYQE+Fgdy+1YVWDGAbPtr2cD4+tqZIxZBRSe5TP+ATwJaI+mUj/h7+3J76/tw/z7L8RD\nhFv//T3/t3AbxW5wNpOaXcS1r69mw4Gj/O2m/vz+2j76EKVFrDrqHYwxh+2vc4AOjdlYRMYB2caY\nLQ1oO0VEUkQkJS9Ph9dQ7mVwTAhfPHoJ914Uw9x1B7hvTgqLNme75AgAxhgWbMzixplrqDaGjx+4\nkBsTO1sdy62Jo/6hichXQMc6Vk0DZhtj2tVqe9QYc7Z+mGjgM2NMvH3ZH1gBXGGMKRKRfUCSMabe\nawBJSUkmJSWlsT+KUi5hW9Yxnlqwje2HjjMoOphnr+lDfGSQ1bGaRfaxUzyzMJVtWUX0jwpi+g39\nCA3US2LNqEmDtDmswJxzpyK7gBHGmMMi0glYaYzpeZa20ZxZYPpS029TYm/SGTgEDDbG5Jxrv1pg\nlLurqjbMTznIi0t3cbSknLuHR3PPRd2IaKW37VZVG95ds4+X/rcLgCeu6MnkYdF42HTQymbWpANq\n1exFi4FJwHT790UN3dAYsw0IP73cmDMYpdydh02YOLgLY/t24tXl6aTsK2TO2hXcmNiZqSNiW9XD\nh9sPFfH0gm1szSrisp5hPDc+ns7BrSe/O7DqDKY9MB/oAuwHbjbGFIpIBPBvY8xYe7v/ACOAUOAI\n8KwxZtZPPmsfeolMqSbJPnaKmSsz+TD5IFXGMD4hkocu6063sECro51VZl4x/1yRya6c4+QcL+P3\n1/bmF3076VD7jtV6LpFZRQuMUnXLKSrlzVWZfPD9ASqqqrl9SFcu79OB4d1DsTnJ5aa0w8eZsSKD\nz7cdxsfTxoOXdmfysBiC/HUk5BagBaY+WmCUOre8E2XMWbuPeckHyTtRRmQ7P25I7MxNiZ0tuXxW\nWVXNmsx83lt3gGU7jhDo48kdF3blnotitBO/ZWmBqY8WGKUaprSiimU7jjA/5SDfZeRjDIxPiCCu\nQxuGdW9P38ggPB30bIkxhi1ZRSzclM1nWw+TX1zGiJ5hDIgKZvKwaD1jsYYWmPpogVGq8bKPneKT\nDVms3VPA2sya8c3a+HgypFt7hse258Lu7YkLC8SjiQWnutqwO/cEyfuOkry3kP0FJ9mSVYS3h42R\nF4QzfkAEI3qG4+vl0Zw/lmocLTD10QKj1PkpKC5j7Z4CVmcUsDojnwOFJXQK8qWguJzIYD86B/sR\nFeJPVLA/Ee18sdXqeBcBYyC/uIxDx06RfewU2cdKMcawNasIgPA2PiRFB3NF745cdkE4QX56tuIk\ntMDUZ8yYMebLL7+0OoZSLuNgYQnbsovYknWMrMJTHDxawsHCEo6WVDA4OoT1+84c6alnh0B2HSnG\nx9NGZLAfke38GNQ1mMhgfwZFhxAV4qd3gzknLTAN4FY/rFJWKS6rpKC4jIqqaqDmzAVqnsMJ8vMi\nJMBbC0nrogWmAdzqh1VKqWbSpAKjQ4wqpZRyCC0wSimlHEILjFJKKYfQAqOUUsohtMAopZRyCC0w\nSimlHEILjFJKKYewasIxq+iTXUop1UL0DEYppZRDaIFRSinlEFpglFJKOYQWGKWUUg6hBUYppZRD\naIFRSinlEFpglFJKOYQWGKWUUg6hBUYppZRDaIFRSinlEP8fJCc1LOrOzicAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9a2d908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0xa05e828>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot (h4.subs('L',1),(x,0,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Funciones de forma en coordenadas naturales\n",
    "\n",
    "En coordenadas naturales $\\xi$, con $\\xi \\in [-1,1]$ las funciones de forma pueden definirse de forma análoga. \n",
    "\n",
    "La base en esta ocasión está formada por las funciones $\\lbrace1,\\xi,\\xi^2,\\xi^3 \\rbrace$.\n",
    "\n",
    "En esta ocasión se evalúa en los extremos del intervalo. \n",
    "\n",
    "Estas funciones pueden obtenerse mediante un procedimiento análogo al visto anteriormente, teniendo en cuenta que se debe evaluar en los extremos $\\xi = -1$ y $\\xi = 1$. \n",
    "\n",
    "Para trabajar en simbólico se establecerá el símbolo $t$ para la letra $\\xi$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAaBAMAAACHnSXKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iVO8Qq5l2zWaJ\nRDIVUIeJAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABRklEQVQ4EWNgIALwNLahq+KZUQYWElIyQJfC\n4B9mqEIX62ZoZOBV0mFQQJfAwldmeI9uRw/DfqBCc6K06zLYX8AwtZFo7QwM/gno2tni4NpZwlAk\n2ZSRuWCeJrIIA0ho1gSY9vT2byiyLChcEI+nAVMBwvGcKOoZOBYgKwbxdiALMEAU2DvAgg5NO9MB\nZNVAHu8B3geoQkCf22/Arv26vDjIX1AA4r1ycUyA8YE0SEiZYSpQCBJxaLb7IykFBjoDQ/z//+hC\nb+YUA4Wwau9CUYvKA0vBhbBq10HRjsoDS8GFsGln+4usHZUHlkEIYdPO+4EhAWEAKg8sjhDCpp1n\nAcsDhHZUHlgcIYRNO0dBJkI3AyoPLIEQAmvnKP1VjaSBWRqYIOAAlQcWRghBbIerJZUxqp3UEIOr\npzjolIgo5+G2oTJ4ldQBeZdPoeUBpLoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & t & t^{2} & t^{3}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡       2   3⎤\n",
       "⎣1  t  t   t ⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = Symbol('t')\n",
    "base_nat = Matrix([[1, t, t**2, t**3]])\n",
    "base_nat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIoAAAAaBAMAAACaz9kUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 1 & 2 t & 3 t^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡              2⎤\n",
       "⎣0  1  2⋅t  3⋅t ⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Dbase_nat = base_nat.diff(t)\n",
    "Dbase_nat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKgAAABkCAMAAAAlv4tJAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADFBd4eAAAAFHRS\nTlMAMquZdlQQQO0wRO/NZt2JIrt8bAICnK4AAAAJcEhZcwAADsQAAA7EAZUrDhsAAAOaSURBVHgB\n7VzZkqQgEKQF6VkvdJb//9blmLIpPKh2cG0j8GEUSYo0QXHKjGYP7baKfeg2eH6MPTQXZqs/lCfr\nLbtKW6KPT+X44tXflWjdytdVpI+IcCIs7i9shhSVFa80nSgRToTFLKNmiKjBijeI0uFvRgXOYbNC\nFFRx+1AaVLFfCJsVRZFWoTSoYr8QNntHUdkN89b5NSIMtdMpBZaI/g7RFSYUBqYZERZ3EDYrRJE6\noTSoYr8QNruporxqdVf1+9f5qiXCibBXXH+Em8WKxuiPKReiuYeiKFoUza1A7nhljhZFcyuQOx6e\no5KrXp2WNhk5n6Z09FEpNS3yS5hoZ+rlMKbVCFMDabRHSJuFU7pJ4EduALWO34wQ0b61QVSXCBWl\nBhLoubpyqY3W9TGfXB4IbdXUQ1SDiE4u81gTchDhK20UcbPYur6rVEauaY3mUsdqIaLaEW3cJW32\n5yqOEO2cAEmiLv7+0EttpwdrtHDgvT9HiPp4A2G8zH2i4t5DRUftqh9+FyNR+TDRhhB8VANfpOow\nUafoqUQHWgJe2YmKtpBo5qFfySgw7qRADNYLbZyntUSf+unR/maqT7qZTB+CwHN0j/EpJvEdpsan\nyRLuCdP92BytLU9PxHa0vmnX/YJoOPTMP/B5/AhbCXiI6MPpKRILX+vUGnQEQ0TZYJfQNr0cH0om\njYNZxHkVLzmxDL1dPOvFIxITlcp80EnzxKmBuKetcue/a6WIsoabl5f4pjcTsnxn2lL22Pmi6DHd\ntlsVRbe1OVZzK0WlWLxUHbvqM1uN5iW5fK/PqfCt5ugth56YWSDCYOwzwNHQEzMLRBiwzANHRE1o\n4hsxEQZcM8ALURDT7YuiSI61QgaJ1sL+nFuLfqs5+vz6M1/e2sXMla8DIgwaZID//cIvJcSQRFhG\norca+nCtJ0pFhJ2mKDGzQIQBzSzweOgh+MftC9HcQ1IULYrmViB3vDJHi6K5FcgdD8/RcxwQROcD\nXNplDgii82Hmab9G7X8G//kgltsBQXQ+ANHrHBBE5wMQvc4B8Y7zAdjuD33mz+DQqd+TnA/Q5EIH\nBMX5ADQvdUAQnQ/A9b86IKBTuyc7H6DRqgNi/i/0NAcExfkAFDccEGhlOssBQXI+ANELHRA05wMQ\nvc4BQXQ+ANHrHBBU5wMwLQ4IUOLcPbrrz+3qd9EL0d/pt2x9M0Vv8osF0v4igBAL89ZS/2vOuF8s\nEIL9AyZTO3tnxMMUAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & -1 & 1 & -1\\\\0 & 1 & -2 & 3\\\\1 & 1 & 1 & 1\\\\0 & 1 & 2 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  -1  1   -1⎤\n",
       "⎢             ⎥\n",
       "⎢0  1   -2  3 ⎥\n",
       "⎢             ⎥\n",
       "⎢1  1   1   1 ⎥\n",
       "⎢             ⎥\n",
       "⎣0  1   2   3 ⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matM_nat = Matrix.vstack(base_nat.subs(t,-1),Dbase_nat.subs(t,-1),base_nat.subs(t,1),Dbase_nat.subs(t,1))\n",
    "matM_nat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAABmCAMAAACjrM68AAAAVFBMVEX///8AAAAAAAAAAAAAAAAA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\n/9NW17mfz7liPuRRCDEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{1}{2} & \\frac{1}{4} & \\frac{1}{2} & - \\frac{1}{4}\\\\- \\frac{3}{4} & - \\frac{1}{4} & \\frac{3}{4} & - \\frac{1}{4}\\\\0 & - \\frac{1}{4} & 0 & \\frac{1}{4}\\\\\\frac{1}{4} & \\frac{1}{4} & - \\frac{1}{4} & \\frac{1}{4}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1/2   1/4   1/2   -1/4⎤\n",
       "⎢                      ⎥\n",
       "⎢-3/4  -1/4  3/4   -1/4⎥\n",
       "⎢                      ⎥\n",
       "⎢ 0    -1/4   0    1/4 ⎥\n",
       "⎢                      ⎥\n",
       "⎣1/4   1/4   -1/4  1/4 ⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Inv_matM_nat = matM_nat.inv()\n",
    "Inv_matM_nat"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Expresión de las funciones de forma en coordenadas naturales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "N1 = base_nat.dot(Inv_matM_nat.col(0))\n",
    "N2 = base_nat.dot(Inv_matM_nat.col(1))\n",
    "N3 = base_nat.dot(Inv_matM_nat.col(2))\n",
    "N4 = base_nat.dot(Inv_matM_nat.col(3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAAwBAMAAAALGvI1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{t^{3}}{4} - \\frac{3 t}{4} + \\frac{1}{2}$$"
      ],
      "text/plain": [
       " 3          \n",
       "t    3⋅t   1\n",
       "── - ─── + ─\n",
       "4     4    2"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJIAAAAwBAMAAAAIgT5YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{t^{3}}{4} - \\frac{t^{2}}{4} - \\frac{t}{4} + \\frac{1}{4}$$"
      ],
      "text/plain": [
       " 3    2        \n",
       "t    t    t   1\n",
       "── - ── - ─ + ─\n",
       "4    4    4   4"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAwBAMAAAAhqgOaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/cN2sPoNVKSpix/eFlQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- \\frac{t^{3}}{4} + \\frac{3 t}{4} + \\frac{1}{2}$$"
      ],
      "text/plain": [
       "   3          \n",
       "  t    3⋅t   1\n",
       "- ── + ─── + ─\n",
       "  4     4    2"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJIAAAAwBAMAAAAIgT5YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\frac{t^{3}}{4} + \\frac{t^{2}}{4} - \\frac{t}{4} - \\frac{1}{4}$$"
      ],
      "text/plain": [
       " 3    2        \n",
       "t    t    t   1\n",
       "── + ── - ─ - ─\n",
       "4    4    4   4"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Nótese que las funciones $N_2(t)$ y $N_4(t)$ proporcionan un giro unitario en la variable $t$, pero no en la variable $x$. \n",
    "\n",
    "Sustituyendo la variable $t$ por $\\xi$, podemos escribir las cuatro funciones de forma como sigue: \n",
    "\n",
    "\\begin{align}\n",
    "N_1(\\xi) & = \\frac{\\xi^{3}}{4} - \\frac{3 \\xi}{4} + \\frac{1}{2} \\\\\n",
    "N_2(\\xi) & = \\frac{\\xi^{3}}{4} - \\frac{\\xi^{2}}{4} - \\frac{\\xi}{4} + \\frac{1}{4} \\\\\n",
    "N_3(\\xi) & = -\\frac{\\xi^{3}}{4} + \\frac{3 \\xi}{4} + \\frac{1}{2} \\\\\n",
    "N_4(\\xi) & =  \\frac{\\xi^{3}}{4} + \\frac{\\xi^{2}}{4} - \\frac{\\xi}{4} - \\frac{1}{4}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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aMqYt3klFZbXV0eQaaCqkNEqaCnl9jp06w6QF25n2/W7iWgfwj2FdiW0dYHUs\nV6WpkCJSP/y9PXjurjimPdqDomOnGDJ5CVMW7qCySkfxzkrlLiLX7I74Vnz9q5u5I74Vr87L5f60\nZewoOm51LLkElbuIXJcgP0+mPNSdScO7safkBHdN/J7py/J04pOTUbmLSK0M7tqGr3/Vj/6dQkhb\ntJOhU7PYeqDM6lhyjspdRGotxN+bqQ9359k7O7Gv9CR3T1zCa1/ncrqyyupojZ7KXURuiGEY3JMY\nyje/vpnBXdsw8dsd3DVxiebFW0zlLiJ1IsjPk38+mMj7P0+ivKKKB9KW8cLcTRzTNz5ZQuUuInXq\nlpgQ5v2qH4/1bsfmgjJue20xc9cX6gtB6pnKXUTqXBMvd54fEs/vBsUS7O/F2FlrGPHuCnYVa9pk\nfdEZqtIo6QzV+lNVbTIzew+vZuZyurKaUTdHMqZ/FN4eblZHc1Z1coaqe11sRETkctxsBiN6t2Ng\n51a8nLGVid/uYNP+Mh7oHsbAzq0wjDrpMvkRDcuISL0I8ffmnw8m8uHTvcg/XM7omatJmbacTYVH\nrY7mkjQsI42ShmWsVVlVzZyV+/jH17kcKT/Dg/Zw/ueOGIL9vayO5gzq5J8yKndplFTuzuFo+Rkm\nLtjOB1l5eHu4Me6nUTzepx1e7o16PF7lLlJbKnfnsrP4OH/9cgvHT1VScKScX90ezX3dQnGzNcrx\neJW7SG2p3J3T0h3FvJKZy/r8o8S09Oc3A2K4NTaksX3oqnIXqS2Vu/MyTZOMDQf4+9e57D50Anvb\nZjw7MIak9o3mu3FV7iK1pXJ3fmeqqvkwZx9TF+3gdKVJTEt/xt/WkaR2QVZHczSVu0htqdwbjpOn\nK5m1Yi9pi3dy6HgFfaNauHrJq9xFakvl3vCUV1QxM3sPaYt34mYYRAY3IfXmSG6JDna1MXmVu0ht\nqdwbrvKKKj5bW8DEBdvZf/QUnVr58/TNkdzdpQ0ebi5xXqbKXaS2VO4NX0VlNZ+vK+TNxTvZXnSc\n0KY+jOnfgcFd2xDg7WF1vBuhchepLZW766iuNlmYW8TURTsoPXGGA2WnGNo9lBG92xHd0t/qeLWh\nchepLZW7a1qff4Tpy/bw+bpCKiqr6R3ZnBG923J7XEvcG86QjcpdpLZU7q6t9EQFH+bsY8ayPRQc\nKSe5Q3MSw5vyQI8wIoObWB3valTuIrWlcm8cqqpNFuYeZHb2PhbmFlFtgr1tM4bZwxmY0MpZx+ZV\n7iK1pXJdZQJ5AAAKUklEQVRvfIrKTvHJmgI+zNlHMx8PNhSW0T8mmMFd2/DTTiH4ejrN11uo3EVq\nS+XeeJmmyfr8o6SvLWTu+kKKjp3Gx8ONlJ7hJIY35ZboEAJ9LT2iV7mL1JbKXeDssM3KvFK+WFfI\nmr1H2Ly/DDebQc92QdwaG8JPO4XQvoVffZ8kpXIXqS2Vu/xYdbXJ2vwjfLP5IAu2FJF78BgdQ5pw\n4nQlvTu0ILlDc3p3aE6bpj6OjqJyl8YhMzOT8ePHU1VVxZNPPsnvfve7C5abpsn48ePJyMjA19eX\n999/n+7du19xmyp3uZq9JSf5fkcxWTtKWLarhNITFQD0aNuUts39SAgNpEtYIHGtA/HxrNMvF1G5\ni+urqqoiOjqa+fPnExYWRlJSErNnzyYuLq5mnYyMDCZNmkRGRgbZ2dmMHz+e7OzsK25X5S7Xo7ra\nJPfgMbJ2lrC39CQZG/ZTfOw0ADYDOob4kxAWSI+IprRp5kv75n60aepd27n1dVLuTvPxsMilrFix\ngqioKCIjIwFISUkhPT39gnJPT09nxIgRGIZBr169OHLkCPv376d169ZWxRYXY7MZxLYOILZ1AAB/\nHhLPwbJTrM8/yoaCo2zIP8Ki3CJ2HzrBqj2HAXC3GYQH+dKuuS+dQwPxdLPRvIkXLZp40ryJF0F+\nHvh4uOPlbsPLw4aXu1udfvOUyl2cWkFBAeHh4TW3w8LCLjoqv9Q6BQUFKndxqJYB3twe583tcS2B\ns8ODRcdOs6fkJHmHTpBXcoI9JSfZfegE5WdKWb6r9ILH92jbrOYvgh+42wx2vDSoTvI5dFgmPj7e\n9PFx+IcPN6S4uJjg4GCrY1xVY815+PBhysrKaNu2LQAlJSWcOHGCiIiImnV27NhBq1ataNLk7JmH\n27ZtIywsDF9f34uyHTp0CIAzZ87QpUuXOsvpKI31dXcUK3OaJlRWV1NZZVJZbVJtmlSd+69pnl1e\njUnB9k2bTNPsXAdPaDrsp0ePHqazawgZTbPx5szKyjLvuOOOmtsvvfSS+dJLL12wTmpqqjlr1qya\n29HR0WZhYWG95nQU5axbDSEnkGPWQf82mCvpSOOUlJTE9u3b2b17NxUVFcyZM4chQ4ZcsM6QIUOY\nPn06pmmyfPlyAgMDNSQjjZ7G3MWpubu7M3nyZAYMGEBVVRUjR44kPj6etLQ0AEaNGsWgQYPIyMgg\nKioKX19f3nvvPYtTi1jPoeWemprqyM3XiYaQERp3zkGDBjFo0IUfMo0aNarmd8MwmDJlynVtszHv\nT0dQzjo1rS42onnuIiLOpU7mQ2rMXUTEBd1QuRuG8TPDMDYZhlFtGIb9cutlZmYSExNDVFQUEyZM\nqLm/tLSU22+/nY4dO3L77bdz+PDhy23ihlzL8+Tm5pKYmFjzExAQwL/+9S8Ann/+eUJDQ2uWZWRk\nWJYToF27diQkJJCYmIjdbr/ux9dHzn379tG/f3/i4uKIj4/n9ddfr1nmyP15uffaD0zT5Be/+AVR\nUVF06dKF1atXX/Nj69LVnmvmzJl06dKFhIQEkpOTWbduXc2yy73+VuRctGgRgYGBNa/lCy+8cM2P\nrc+cr776ak3Gzp074+bmRmnp2Xnn9bU/R44cSUhICJ07X3qW4w/vTcMwdhiGsd4wjJpraBiGMdAw\njNxzy353yQ1caoO1/QFigRhgEWC/xDpmZWWlGRkZae7cudM8ffq02aVLF3PTpk2maZrmb37zG/Pl\nl182TdM0X375ZfO3v/1t3c8rqsXzVFZWmi1btjTz8vJM0zTNP/3pT+arr77qkGy1ydm2bVuzuLi4\n1o+vj5yFhYXmqlWrTNM0zbKyMrNjx441r7uj9ueV3ms/+PLLL82BAwea1dXV5rJly8yePXte82Pr\nM+fSpUvN0tJS0zRNMyMjoyanaV7+9bci58KFC8277rqrVo+tz5zn+/zzz83+/fvX3K6v/bl48WJz\n1apVZnx8/CWX//De5OywTC8g2zzbs27ATiAS8ATWAXGmI6dCmqa5xTTN3Cutc/7p456enjWnj8PZ\n08Yfe+wxAB577DE+++yzG4lzWdf7PAsWLKBDhw41J87UlxvdH860P1u3bl1z8S5/f39iY2MpKChw\nSJ4fXOm9dn72S12q4FoeW585k5OTadasGQC9evUiPz/fIVluNKcjHuvonLNnz2b48OEOyXIl/fr1\nIygo6LLLf3hvnuv65UBTwzBaAz2BHaZp7jJNswKYA9xz1Se8Wvtfyw+XP3IHeAB4+7zbjwKTz/1+\n5Lz7jfNv1+XP9T4P8C4w9rzbzwN7gPXnljWzMiewG1gLrAJSa/vnrMf92Q7YCwQ4cn9e6b123n1z\ngb7n3V4A2K/lsXW4/67ruYD//dH6l3z9rcgJ3AKUnnstvwLia/NnrK/9CfieyxtU3/vz3HO1AzZe\nZlmdvjevOhXSMIxvgFaXWPQH0zTr7K9i0zRNwzBqPbvmSjmv53kMw/AEhgC/P+/uqcCLnJ398yLw\nD2CkhTn7mqZZYBhGCDDfMIytpml+dx2Pr6+cGIbRBPgY+KVpmmXn7q6z/enqDMPoDzwB9D3v7qu+\n/vVoNRBhmuZxwzAGAZ8BHS3Kci0GA0tN0zz/Qi/OtD/rzFXL3TTN227wOQqA8PNuh527D+CgYRit\nTdPcf+6fH0W1fZIr5TQM43qe505gtWmaB8/bds3vhmG8xdm/YS3LaZpmwbn/FhmG8Sln/9n2HU62\nPw3D8OBssc80TfOT87ZdZ/vzR670XrvaOh7X8Ni6ci05MQyjC/A2cKdpmiU/3H+F17/ec573Fzam\naWYYhvGGYRgtruWx9ZnzPCnA7PPvqMf9eTV1+t6sj6mQK4GOhmG0P3dUnAJ8fm7Z58Bj535/DHDM\noNz1Pc9wfvTinyuwH9wHbKzTdP/fVXMahuFnGIb/D78Dd5yXx2n2p3H2e8neAbaYpvnaj5Y5an9e\n6b12fvYRxlm9gKOmae6/xsfWlas+l2EYEcAnwKOmaW477/4rvf5W5Gx17rXGMIyenO2Ukmt5bH3m\nPJcvELiZ896v9bw/r6Zu35s3OH50H5APnAYOAvPO3d8GyDhvvUHANs5+4vuH8+5vztlxpe3AN5w3\nDlbH41yXfJ5L5PTj7Bsz8EePnwFs4Oy44udAa6tycvYT83XnfjY56/7k7DCCeW6frT33M8jR+/NS\n7zVgFDDq3O8GMOXc8g2c91nR5d6nDtqHV8v5NnD4vH2Xc7XX36KcY8/lWAcsB5KdcX+eu/04MOdH\nj6u3/cnZg8b9wBnO9uYTjnxvOvoMVRERsYDOUBURcUEqdxERF6RyFxFxQSp3EREXpHIXEXFBKncR\nESdhGEZTwzDG1MW2VO4iIs6jKaByFxFxMROADoZhrDUM49Ub2ZBOYhIRcRKGYbQD5pqmeelv9LgO\nOnIXEXFBKncRERekchcRcR7HAP+62JDKXUTESZhnr9u/1DCMjfpAVURELqIjdxERF6RyFxFxQSp3\nEREXpHIXEXFBKncRERekchcRcUEqdxERF6RyFxFxQf8Ps7no4KYBCKUAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9b370b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0xa06d710>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot (N1,(t,-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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zgUFU1DXy5NLdNBtbtI4kTMDmyn1HfgUVtY1MiO6mdRQh7Ea/IB+eHduX7Ucq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\nAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xaf6a550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0xaf6a7b8>"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot (N2,(t,-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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qqut4fsl2Zn69m1bN/Xj6+j78YkAUTXzomjAiYi+VuxsZY1i86Tse+2ATBw5V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\nIj5I5S4i4oNU7iIiPkjlLiLig/4fXsWZGXi637gAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb097be0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0xb16e080>"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot (N3,(t,-1,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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8mhR3AUAfz07ETAvmcHoR/9pySus4Qmhuy4ls7KytTHoR7OZIcRf1JoV045Hh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\nhRDCAklxF0IICyTFXQghLND/B0/049MvlY4CAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9b2a4e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0xb31c240>"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot (N4,(t,-1,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ejercicios\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Ejercicio 1\n",
    "Determine las funciones de forma asociadas a una base lagrangiana de polinomios de grado 4, asociada a los 5 puntos $x= \\lbrace 0, \\dfrac{L}{4}, \\dfrac{L}{2}, \\dfrac{3L}{4}, L \\rbrace$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solución \n",
    "\n",
    "Consideremos los puntos de evaluación. Procediendo según se vio anteriormente, se obtiene la matriz $\\mathbf{M}$ y su inversa. En esta base lagrangiana, la función de forma de índice $i$ debe tomar valor 1 en el punto $i$, y cero en el resto. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "puntos = [0, L/4, L/2, 3*L/4, L]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALIAAAAaBAMAAADh6xvYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & x & x^{2} & x^{3} & x^{4}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡       2   3   4⎤\n",
       "⎣1  x  x   x   x ⎦"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "base = Matrix([[1,x,x**2,x**3,x**4]])\n",
    "base"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPkAAACJCAMAAADZs25jAAAAZlBMVEX///8AAAAAAAAAAAAAAAAA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\nYYPX32A27XDw+ztZiyXc/AsDgN1h6DulkAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0 & 0 & 0\\\\1 & \\frac{L}{4} & \\frac{L^{2}}{16} & \\frac{L^{3}}{64} & \\frac{L^{4}}{256}\\\\1 & \\frac{L}{2} & \\frac{L^{2}}{4} & \\frac{L^{3}}{8} & \\frac{L^{4}}{16}\\\\1 & \\frac{3 L}{4} & \\frac{9 L^{2}}{16} & \\frac{27 L^{3}}{64} & \\frac{81 L^{4}}{256}\\\\1 & L & L^{2} & L^{3} & L^{4}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   0    0      0      0  ⎤\n",
       "⎢                          ⎥\n",
       "⎢          2     3       4 ⎥\n",
       "⎢    L    L     L       L  ⎥\n",
       "⎢1   ─    ──    ──     ─── ⎥\n",
       "⎢    4    16    64     256 ⎥\n",
       "⎢                          ⎥\n",
       "⎢          2     3      4  ⎥\n",
       "⎢    L    L     L      L   ⎥\n",
       "⎢1   ─    ──    ──     ──  ⎥\n",
       "⎢    2    4     8      16  ⎥\n",
       "⎢                          ⎥\n",
       "⎢           2      3      4⎥\n",
       "⎢   3⋅L  9⋅L   27⋅L   81⋅L ⎥\n",
       "⎢1  ───  ────  ─────  ─────⎥\n",
       "⎢    4    16     64    256 ⎥\n",
       "⎢                          ⎥\n",
       "⎢          2     3      4  ⎥\n",
       "⎣1   L    L     L      L   ⎦"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matM = Matrix.vstack(base.subs(x,puntos[0]), base.subs(x,puntos[1]), base.subs(x,puntos[2]), base.subs(x,puntos[3]),\n",
    "                    base.subs(x,puntos[4]))\n",
    "matM"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUwAAAB9CAMAAAD+4arRAAAAhFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0 & 0 & 0\\\\- \\frac{25}{3 L} & \\frac{16}{L} & - \\frac{12}{L} & \\frac{16}{3 L} & - \\frac{1}{L}\\\\\\frac{70}{3 L^{2}} & - \\frac{208}{3 L^{2}} & \\frac{76}{L^{2}} & - \\frac{112}{3 L^{2}} & \\frac{22}{3 L^{2}}\\\\- \\frac{80}{3 L^{3}} & \\frac{96}{L^{3}} & - \\frac{128}{L^{3}} & \\frac{224}{3 L^{3}} & - \\frac{16}{L^{3}}\\\\\\frac{32}{3 L^{4}} & - \\frac{128}{3 L^{4}} & \\frac{64}{L^{4}} & - \\frac{128}{3 L^{4}} & \\frac{32}{3 L^{4}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 1      0      0      0     0  ⎤\n",
       "⎢                               ⎥\n",
       "⎢-25    16    -12      16   -1  ⎥\n",
       "⎢────   ──    ────    ───   ─── ⎥\n",
       "⎢3⋅L    L      L      3⋅L    L  ⎥\n",
       "⎢                               ⎥\n",
       "⎢ 70   -208    76    -112    22 ⎥\n",
       "⎢────  ─────   ──    ─────  ────⎥\n",
       "⎢   2      2    2        2     2⎥\n",
       "⎢3⋅L    3⋅L    L      3⋅L   3⋅L ⎥\n",
       "⎢                               ⎥\n",
       "⎢-80    96    -128   224    -16 ⎥\n",
       "⎢────   ──    ─────  ────   ────⎥\n",
       "⎢   3    3       3      3     3 ⎥\n",
       "⎢3⋅L    L       L    3⋅L     L  ⎥\n",
       "⎢                               ⎥\n",
       "⎢ 32   -128    64    -128    32 ⎥\n",
       "⎢────  ─────   ──    ─────  ────⎥\n",
       "⎢   4      4    4        4     4⎥\n",
       "⎣3⋅L    3⋅L    L      3⋅L   3⋅L ⎦"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Inv_matM = matM.inv()\n",
    "Inv_matM"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f1 = base.dot(Inv_matM.col(0))\n",
    "f2 = base.dot(Inv_matM.col(1))\n",
    "f3 = base.dot(Inv_matM.col(2))\n",
    "f4 = base.dot(Inv_matM.col(3))\n",
    "f5 = base.dot(Inv_matM.col(4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuación se muestran las funciones de forma; para una mejor visualización se define un vector y se muestran por columnas. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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YtlJuLVJKRxjntb4XXAVVONJTbsG6Toq2xbnaZfGCsG2m3Bp6eOnInnLrIYLXZcotGNVJ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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 - \\frac{25 x}{3 L} + \\frac{70 x^{2}}{3 L^{2}} - \\frac{80 x^{3}}{3 L^{3}} + \\frac{32 x^{4}}{3 L^{4}}\\\\\\frac{16 x}{L} - \\frac{208 x^{2}}{3 L^{2}} + \\frac{96 x^{3}}{L^{3}} - \\frac{128 x^{4}}{3 L^{4}}\\\\- \\frac{12 x}{L} + \\frac{76 x^{2}}{L^{2}} - \\frac{128 x^{3}}{L^{3}} + \\frac{64 x^{4}}{L^{4}}\\\\\\frac{16 x}{3 L} - \\frac{112 x^{2}}{3 L^{2}} + \\frac{224 x^{3}}{3 L^{3}} - \\frac{128 x^{4}}{3 L^{4}}\\\\- \\frac{x}{L} + \\frac{22 x^{2}}{3 L^{2}} - \\frac{16 x^{3}}{L^{3}} + \\frac{32 x^{4}}{3 L^{4}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡               2       3       4⎤\n",
       "⎢    25⋅x   70⋅x    80⋅x    32⋅x ⎥\n",
       "⎢1 - ──── + ───── - ───── + ─────⎥\n",
       "⎢    3⋅L        2       3       4⎥\n",
       "⎢            3⋅L     3⋅L     3⋅L ⎥\n",
       "⎢                                ⎥\n",
       "⎢             2       3        4 ⎥\n",
       "⎢ 16⋅x   208⋅x    96⋅x    128⋅x  ⎥\n",
       "⎢ ──── - ────── + ───── - ────── ⎥\n",
       "⎢  L         2       3        4  ⎥\n",
       "⎢         3⋅L       L      3⋅L   ⎥\n",
       "⎢                                ⎥\n",
       "⎢             2        3       4 ⎥\n",
       "⎢  12⋅x   76⋅x    128⋅x    64⋅x  ⎥\n",
       "⎢- ──── + ───── - ────── + ───── ⎥\n",
       "⎢   L        2       3        4  ⎥\n",
       "⎢           L       L        L   ⎥\n",
       "⎢                                ⎥\n",
       "⎢            2        3        4 ⎥\n",
       "⎢16⋅x   112⋅x    224⋅x    128⋅x  ⎥\n",
       "⎢──── - ────── + ────── - ────── ⎥\n",
       "⎢3⋅L        2        3        4  ⎥\n",
       "⎢        3⋅L      3⋅L      3⋅L   ⎥\n",
       "⎢                                ⎥\n",
       "⎢            2       3       4   ⎥\n",
       "⎢    x   22⋅x    16⋅x    32⋅x    ⎥\n",
       "⎢  - ─ + ───── - ───── + ─────   ⎥\n",
       "⎢    L       2      3        4   ⎥\n",
       "⎣         3⋅L      L      3⋅L    ⎦"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fforma = Matrix([f1,f2,f3,f4,f5])\n",
    "fforma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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PT+PaRRNI9vs6BnvcSAbFemCqmZUR/vC/Dvj0Kfs8CdxuZo8S7pZq7Gt8QoZf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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb32c710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0xba48be0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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cP7Qn1wzsxis7df4tV2HlGckoINUYk2aMqQFeA2Zf0GY2sNo02AF0FhEdxOAk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\n8/jmyCh6hFxxRwylPFLfbh2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      "text/plain": [
       "<matplotlib.figure.Figure at 0xbe6bdd8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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EhQTw0/N1YkZ1an1jQ6lqaKGqvpmo0ACHHNOyQiIivsDTwPlAEbBRRJYaY7Z3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\nNR5YbzrxneypLZJTTq/y7rv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      "text/plain": [
       "<matplotlib.figure.Figure at 0xbea4828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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+mSne9oWXzKzrXN/rZVCsB6aaWSWhg/8K4HMD2jwH3B3uv7gYaBlO/8Qn55ex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\nJAWFiIgMSUEhIiJD+v9Xg+3hhqOdEAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb324d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0xbe92f60>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(f1.subs(L,1),(x,0,1),title=\"$f_1(x)$\")\n",
    "plot(f2.subs(L,1),(x,0,1),title=\"$f_2(x)$\")\n",
    "plot(f3.subs(L,1),(x,0,1),title=\"$f_3(x)$\")\n",
    "plot(f4.subs(L,1),(x,0,1),title=\"$f_4(x)$\")\n",
    "plot(f5.subs(L,1),(x,0,1),title=\"$f_5(x)$\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ejercicio 2\n",
    "Determine las funciones de forma asociadas a una base de polinomios generados por la base $\\mathcal{B} = \\lbrace{1,x,x^2, x^4} \\rbrace$ , con las mismas condiciones que el elemento hermítico visto anteriormente (desplazamientos y giros en los extremos del intervalo $[0,L]$).\n",
    "\n",
    "Observación: Esta base representa una base incompleta de los polinomios de grado 4, pues falta el término $x^3$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solución\n",
    "\n",
    "Procediendo según lo visto anteriormente, se obtienen las cuatro funciones de forma. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAABzCAMAAAB5L8r9AAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 - \\frac{2 x^{2}}{L^{2}} + \\frac{x^{4}}{L^{4}}\\\\x - \\frac{3 x^{2}}{2 L} + \\frac{x^{4}}{2 L^{3}}\\\\\\frac{2 x^{2}}{L^{2}} - \\frac{x^{4}}{L^{4}}\\\\- \\frac{x^{2}}{2 L} + \\frac{x^{4}}{2 L^{3}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡        2    4 ⎤\n",
       "⎢     2⋅x    x  ⎥\n",
       "⎢ 1 - ──── + ── ⎥\n",
       "⎢       2     4 ⎥\n",
       "⎢      L     L  ⎥\n",
       "⎢               ⎥\n",
       "⎢       2     4 ⎥\n",
       "⎢    3⋅x     x  ⎥\n",
       "⎢x - ──── + ────⎥\n",
       "⎢    2⋅L       3⎥\n",
       "⎢           2⋅L ⎥\n",
       "⎢               ⎥\n",
       "⎢      2    4   ⎥\n",
       "⎢   2⋅x    x    ⎥\n",
       "⎢   ──── - ──   ⎥\n",
       "⎢     2     4   ⎥\n",
       "⎢    L     L    ⎥\n",
       "⎢               ⎥\n",
       "⎢     2     4   ⎥\n",
       "⎢    x     x    ⎥\n",
       "⎢ - ─── + ────  ⎥\n",
       "⎢   2⋅L      3  ⎥\n",
       "⎣         2⋅L   ⎦"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "base = Matrix([[1,x,x**2,x**4]])\n",
    "Dbase = base.diff(x)\n",
    "matM = Matrix.vstack(base.subs(x,0), Dbase.subs(x,0),base.subs(x,L),Dbase.subs(x,L))\n",
    "Inv_matM = matM.inv()\n",
    "f1 = base.dot(Inv_matM.col(0))\n",
    "f2 = base.dot(Inv_matM.col(1))\n",
    "f3 = base.dot(Inv_matM.col(2))\n",
    "f4 = base.dot(Inv_matM.col(3))\n",
    "\n",
    "fforma = Matrix([f1,f2,f3,f4])\n",
    "\n",
    "fforma\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A continuación se dibujan las funciones de forma. Observe que se parecen mucho a las hermíticas, pero observando la matriz anterior, puede comprobar que no contienen términos en $x^3$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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/28zQSQvYc+SE17EqJQu2qwuSk5NdWpr6RKQy+WDpLl5P3cq2Q9n888YO9G6V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\nEZFiqShERKRYKgoRESmWikJERIr1/wFe3xR2hEbyOQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xba5cef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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/uZjYyDCm3jGEDi0aOV1STdTpi9GvH977UuGRMhZu3sPIPq0tVIwx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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xb2fa278>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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33XVOpIFTUQiFB0t5+tONzPx0I82aNOKWM/py27gEHWYSEUBF0aC5O3NXbOeB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\nSUUhIiJV+v//TvwaF7E5wAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc1bd6a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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5CleO7f7RXZk6NIr569PYlKrT+yvXtzIxmxA/L+I6W/twsbpYFTATgEW214uA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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xc4306d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0xc441d30>"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(f1.subs(L,1),(x,0,1),title=\"$f_1(x)$\")\n",
    "plot(f2.subs(L,1),(x,0,1),title=\"$f_2(x)$\")\n",
    "plot(f3.subs(L,1),(x,0,1),title=\"$f_3(x)$\")\n",
    "plot(f4.subs(L,1),(x,0,1),title=\"$f_4(x)$\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}