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In the galileo shell, you can interact with mathematical expresssions. The examples below illustrate the currently available features.

Basics

Working with numbers

galileo>5
5
galileo>5+6
11

By default, galileo does not evaluate fractions. The 'eval' command can be used to force the evaluation.

galileo>5/4
5/4
galileo>eval(5/4)
1.25

Working with variables

galileo>x+7
x+7
galileo>x=3
3
galileo>x+7
10

Complex numbers

The imaginary unit is available as either i or j, with i*i=-1

galileo>2+3*i
3*j+2
galileo>j*j
-1

Manipulation of expressions

The commands 'simplify', 'factor' and 'expand' can be used to manipulate expressions.

The 'simplify' command simplifies an expression:

m=metric.generate(three-sphere)
et=einsteintensor(m)

The Einstein tensor for any metric contains combinations of second order derivatives of the metric expression, which can be quite complex before simplification:

Tensor(List(TensorIndex(Lower,3), TensorIndex(Lower,3)),List(((-1.0)*(2.0*(2.0*r^2.0*cos(psi)^2.0-2.0*r^2.0*sin(psi)^2.0)*r^2.0*sin(psi)^2.0-8.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0)-4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0)/(4.0*r^4.0*sin(psi)^4.0)+((-1.0)*(2.0*(2.0*r^2.0*cos(psi)^2.0*sin(theta)^2.0-2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0*sin(theta)^2.0-8.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0)-4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0)+(-1.0)*((((-1.0)*(2.0*(2.0*r^2.0*cos(psi)^2.0-2.0*r^2.0*sin(psi)^2.0)*r^2.0*sin(psi)^2.0-8.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0)-4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0)/(4.0*r^4.0*sin(psi)^4.0)+((-1.0)*(2.0*(2.0*r^2.0*cos(psi)^2.0*sin(theta)^2.0-2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0*sin(theta)^2.0-8.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0)-4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0))/r^2.0+(2.0*((-2.0)*r^2.0*cos(psi)^2.0+2.0*r^2.0*sin(psi)^2.0)*r^2.0/(4.0*r^4.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0/(4.0*r^4.0*sin(psi)^2.0)+(-1.0)*(2.0*(2.0*r^2.0*sin(psi)^2.0*cos(theta)^2.0-2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0*sin(theta)^2.0-8.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0))/(r^2.0*sin(psi)^2.0)+(2.0*((-2.0)*r^2.0*cos(psi)^2.0*sin(theta)^2.0+2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0/(4.0*r^4.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+2.0*((-2.0)*r^2.0*sin(psi)^2.0*cos(theta)^2.0+2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0/(4.0*r^4.0*sin(psi)^4.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^2.0)+4.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+4.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0))/(r^2.0*sin(psi)^2.0*sin(theta)^2.0))*r^2.0/2.0, 4.0*r^4.0*sin(psi)^2.0*cos(psi)*sin(psi)*cos(theta)*sin(theta)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^2.0*sin(theta)^2.0*cos(psi)*sin(psi)*cos(theta)*sin(theta)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0), 0.0, 4.0*r^4.0*sin(psi)^2.0*cos(psi)*sin(psi)*cos(theta)*sin(theta)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^2.0*sin(theta)^2.0*cos(psi)*sin(psi)*cos(theta)*sin(theta)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0), 2.0*((-2.0)*r^2.0*cos(psi)^2.0+2.0*r^2.0*sin(psi)^2.0)*r^2.0/(4.0*r^4.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0/(4.0*r^4.0*sin(psi)^2.0)+(-1.0)*(2.0*(2.0*r^2.0*sin(psi)^2.0*cos(theta)^2.0-2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0*sin(theta)^2.0-8.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0)+(-1.0)*r^2.0*sin(psi)^2.0*((((-1.0)*(2.0*(2.0*r^2.0*cos(psi)^2.0-2.0*r^2.0*sin(psi)^2.0)*r^2.0*sin(psi)^2.0-8.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0)-4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0)/(4.0*r^4.0*sin(psi)^4.0)+((-1.0)*(2.0*(2.0*r^2.0*cos(psi)^2.0*sin(theta)^2.0-2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0*sin(theta)^2.0-8.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0)-4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0))/r^2.0+(2.0*((-2.0)*r^2.0*cos(psi)^2.0+2.0*r^2.0*sin(psi)^2.0)*r^2.0/(4.0*r^4.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0/(4.0*r^4.0*sin(psi)^2.0)+(-1.0)*(2.0*(2.0*r^2.0*sin(psi)^2.0*cos(theta)^2.0-2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0*sin(theta)^2.0-8.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0))/(r^2.0*sin(psi)^2.0)+(2.0*((-2.0)*r^2.0*cos(psi)^2.0*sin(theta)^2.0+2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0/(4.0*r^4.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+2.0*((-2.0)*r^2.0*sin(psi)^2.0*cos(theta)^2.0+2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0/(4.0*r^4.0*sin(psi)^4.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^2.0)+4.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+4.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0))/(r^2.0*sin(psi)^2.0*sin(theta)^2.0))/2.0, 0.0, 0.0, 0.0, 2.0*((-2.0)*r^2.0*cos(psi)^2.0*sin(theta)^2.0+2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0/(4.0*r^4.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+2.0*((-2.0)*r^2.0*sin(psi)^2.0*cos(theta)^2.0+2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0/(4.0*r^4.0*sin(psi)^4.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^2.0)+4.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+4.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+(-1.0)*r^2.0*sin(psi)^2.0*((((-1.0)*(2.0*(2.0*r^2.0*cos(psi)^2.0-2.0*r^2.0*sin(psi)^2.0)*r^2.0*sin(psi)^2.0-8.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0)-4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0)/(4.0*r^4.0*sin(psi)^4.0)+((-1.0)*(2.0*(2.0*r^2.0*cos(psi)^2.0*sin(theta)^2.0-2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0*sin(theta)^2.0-8.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0)-4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0))/r^2.0+(2.0*((-2.0)*r^2.0*cos(psi)^2.0+2.0*r^2.0*sin(psi)^2.0)*r^2.0/(4.0*r^4.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0/(4.0*r^4.0*sin(psi)^2.0)+(-1.0)*(2.0*(2.0*r^2.0*sin(psi)^2.0*cos(theta)^2.0-2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0*sin(theta)^2.0-8.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0)/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^4.0))/(r^2.0*sin(psi)^2.0)+(2.0*((-2.0)*r^2.0*cos(psi)^2.0*sin(theta)^2.0+2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0/(4.0*r^4.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+2.0*((-2.0)*r^2.0*sin(psi)^2.0*cos(theta)^2.0+2.0*r^2.0*sin(psi)^2.0*sin(theta)^2.0)*r^2.0*sin(psi)^2.0/(4.0*r^4.0*sin(psi)^4.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^2.0)+4.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+(-4.0)*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+(-4.0)*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0)+4.0*r^4.0*cos(psi)^2.0*sin(psi)^2.0*sin(theta)^4.0/(4.0*r^4.0*sin(psi)^2.0*sin(theta)^2.0)+4.0*r^4.0*sin(psi)^4.0*cos(theta)^2.0*sin(theta)^2.0/(4.0*r^4.0*sin(psi)^4.0*sin(theta)^2.0))/(r^2.0*sin(psi)^2.0*sin(theta)^2.0))*sin(theta)^2.0/2.0))

The 'simplify' command reduces the complexity of this expression:

simplify(einsteintensor(metric.generate(three-sphere)))

returns a mere

Tensor(List(TensorIndex(Lower,3), TensorIndex(Lower,3)),List(-1.0, 0.0, 0.0, 0.0, (-1.0)*sin(psi)^2.0, 0.0, 0.0, 0.0, (-4.0)*sin(psi)^2.0*sin(theta)^2.0/4.0))

Logic operations

galileo>true && false
false
galileo>or(false,true)
true

Basic calculus

You can take the derivative using the 'deriv' command.

galileo>deriv(x^2,x)
2.0*x

Matrices and tensors

Matrix manipulation

The following commands are available:

  • lu(A) LU factorization for matrix A
  • inv(A) Inverse of matrix A
  • rand(n) Random nxn matrix; elements chosen from uniform distribution
  • rand(n,m) Random nxm matrix; elements chosen from uniform distribution
  • eye(n) nxn Identity matrix
  • ones(n) nxn matrix filled with all ones
  • linspace
  • logspace
galileo>A=[a b;c d]
a    b
c    d
galile0>A[0]
a    b
galileo>A[0,1]
b
galileo>[L,U,P]=lu(A)
L	=
1.0	0.0	
c/a	1.0	

U	=
a	b	
0.0	(-1.0)*b*c/a+d	

P	=
1.0	0.0
0.0	1.0
galileo>L*U
a	b
c	d
galileo>P*A
a	b
c	d
galileo>A=[3 4;5 b]
3.0	4.0
5.0	b
galileo>inv(A)
(-1.0)*b/((-3.0)*b+20.0)	3.0*b/(5.0*((-3.0)*b+20.0))+1.0/5.0
5.0/((-3.0)*b+20.0)	(-15.0)/(5.0*((-3.0)*b+20.0))
galileo> simplify(ans)
(-1.0)*b/((-3.0)*b+20.0)	3.0*b/((-15.0)*b+100.0)+1.0/5.0
5.0/((-3.0)*b+20.0)	(-15.0)/((-15.0)*b+100.0)

Tensor manipulation

galileo> m=metric.generate(three-sphere)
Metric(Tensor(List(TensorIndex(Lower,3), TensorIndex(Lower,3)),List(r^2.0, 0.0, 0.0, 0.0, r^2.0*sin(psi)^2.0, 0.0, 0.0, 0.0, r^2.0*sin(psi)^2.0*sin(theta)^2.0)),List(psi, theta, phi ))
galileo> simplify(einsteintensor(m))
Tensor(List(TensorIndex(Lower,3), TensorIndex(Lower,3)),List(-1.0, 0.0, 0.0, 0.0, (-1.0)*sin(psi)^2.0, 0.0, 0.0, 0.0, sin(psi)^2.0*sin(theta)^2.0+(-1.0)*cos(psi)^2.0*sin(theta)^2.0+(-3.0)*sin(psi)^2.0*sin(theta)^2.0+sin(theta)^2.0))