# Overview

• It was when I found out I could make mistakes that I knew I was on to something. (Ornette Coleman)
• Jazz is a mental attitude rather than a style. It uses a certain process of the mind expressed spontaneously through some musical instrument. I’m concerned with retaining that process. (Bill Evans)
• Jam Session} = An informal gathering of musicians to play improvised or unrehearsed music. (Online Jam Session)

The BodyandSoul Sessions help you to master basic tools of Calculus and Linear Algebra in interaction with a computer, including the following topics presented in \hyperref[partleibniz]{Leibniz World of Mathematics}:

1. functions
2. Lipschitz continuity
3. derivatives
4. Fundamental Theorems of Calculus and Linear Algebra,
5. elementary functions
6. geometry in $R^2$ and $R^3$
7. fixed point iteration and Newton’s method
8. time stepping and adaptive error control
10. piecewise linear interpolation
11. finite element programming,

Each Session contains simple Python codes which you can use as templates to get a quick start.

A set of Sessions using Matlab is available as Sessions A-F}.

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A new set based on Python is coming up, including additional Sessions related to the material in

• \hyperref[partworldofde]{Part VII World of Differential Equations}.
• \hyperref[partfem]{Part VIII World of Finite Elements}.

The material covered by the Sessions is summarized in

• \hyperref[chapter1Dcalc]{1D Calculus}
• \hyperref[chaptermultiDcalc]{MultiD Calculus}
• \hyperref[chapterLA]{Geometry and Linear Algebra}.

# Python Code

The Sessions includes writing Python code for

• Functions.
• Computing derivatives, analytically and computationally.
• Time stepping of $\dot u=f(t)$ and $\dot u=f(u)$. Quadrature in 1d.
• Computing elementary functions.
• Computing maximum of a Lipschitz continuous function.
• Fixed point iteration for $x=g(x)$.
• Newton’s method for $f(x)=0$.
• Transformation of matrix to row and column echelon form.
• Gaussian elimination.
• Jacobi iteration for linear system $Ax=b$ with $A$ diagonally dominant.
• Conjugate gradient method for linear system $Ax$ with $A$ positive definite.
• Least squares method for linear system $Ax=b$.
• Geometry in $R^2$ and $R^3$: scalar and vector product, projection, reflection, rotation,..
• Level curves of $f:R^2\rightarrow R$.
• Level surfaces of $f:R^3\rightarrow R$.
• Graphics: curves, surfaces and volumes.
• Quadrature in 2d and 3d.