Levi-Civita symbol and cross product

 

Reference:

http://www.homepages.ucl.ac.uk/~ucappgu/seminars/levi-civita.pdf

https://en.wikipedia.org/wiki/Levi-Civita_symbol

Average over a sphere surface

 The surface integral of the function $f$ over the surface $S$ is denoted by

\[ \int\int_S f dS \]

$dS$ is the area of an infinitesimal piece of the surface $S$.

Average over the sphere is

\[ <f> = \frac{1}{4\pi} \int_0^{2\pi} d\phi \int_0^{\pi} f \sin(\theta)d\theta ,\]

where $dS=\sin(\theta)d\theta d\phi$. This can be seen as a weight average, the weight is surface area.

\[ <\cos^2(\theta)>=1/3 \]

\[<\sin(\theta)>=2/3.\]

A simple guide to Install scikit ODES

Here is a simple guid to install scikit-odes. Odes is a scikit toolkit for scipy to add extra ode solvers. Specifically it interfaces the Sundials solvers cvode, cvodes, ida and idas.

a) Install BLAS and LAPACK

>sudo apt-get install libopenblas-dev liblapack-dev

b) Install Sundials 5.1.0

>wget https://computing.llnl.gov/projects/sundials/download/sundials-5.1.0.tar.gz

>tar xvf sundials-5.1.0.tar.gz 

>mkdir builder 

>cd builder 

>cmake ../../sundials-5.1.0/ -DLAPACK_ENABLE=ON 

>make 

>sudo make install

c) Install scikits odes

add export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:/usr/local/bin to .bashrc 

>source ~/.bashrc 

>sudo pip install scikits.odes

Reference:

https://scikits-odes.readthedocs.io/en/latest/installation.html

Compute Jacobian matrix by Sympy

Computing the Jacobian matrix is not easy, but we can use the python symbolical package Sympy to get it.

Here is a simple example.

import sympy as sym
from sysmpy import init_printing
from sympy import sin,cos,Matrix

init_printing()

x,y=sym.symbols('x y')
F=Matrix([x*cos(y),x*sin(y),x**2])
X=Matrix([x,y])
print(F.jacobian(X))
print(F.jacbian(X).subs([(x,0),(y,1)]))

Reference:

https://docs.sympy.org/latest/modules/matrices/matrices.html

https://www.sympy.org/scipy-2017-codegen-tutorial/notebooks/20-ordinary-differential-equations.html

The solution of diffusion equation

Reference:

http://folk.ntnu.no/leifh/teaching/tkt4140/._main034.html

Relaxation method to solve the differential equation

Reference:

"The second method for numerical solution of BVPs is known as finite differences, relaxation, or -in the specific case of stellar structures- the Henyey method".

! IMPORTANT!

http://dipastro.pd.astro.it/chiosi/Lezioni/LAUREA_TRIENNALE/ASTROFISICA_II_MOD_B/LEZIONE_XIV/Binder_XIV.pdf

Lectures on Numerical Mathematics 

chapter 10 Relaxation Methods 

HENYEY RELAXATION METHOD REVISITED: AN APPLICATION TO PROBLEMS INVOLVING

CRITICAL POINTS

FUNDAMENTALS OF ENGINEERING NUMERICAL ANALYSIS