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Introduction to Linear Algebra

## Glossary

# Preface

This tutorial was made solely for the purpose of education and it was designed
for students taking Applied Math 0340. It is primarily for students who
have some experience using *Mathematica*. If you have never used
*Mathematica* before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA
0330 tutorial. As a friendly reminder, don't forget to clear variables in use and/or the kernel. The *Mathematica* commands in this tutorial are all written in **bold black font**, while *Mathematica* output is in regular fonts.

Finally, you can copy and paste all commands into your *Mathematica* notebook, change the parameters, and run them because the tutorial is under the terms of the GNU General Public License
(GPL). You, as the user, are free to use the scripts for your needs to learn the *Mathematica* program, and have
the right to distribute and refer to this tutorial, as long as
this tutorial is accredited appropriately. The tutorial accompanies the
textbook *Applied Differential Equations.
The Primary Course* by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043

# Poisson equation

It is known that the electric field generated by a set of stationary charges
can be written as the gradient of a scalar potential, so that
**E** = -∇φ. This equation can be combined with the field
equation to give a partial differential equation for the scalar potential:
∇²φ = -ρ/ε_{0}. This is an example of a
very famous type of partial differential equation known as **Poisson's
equation**:

*L*is an elliptic differential operator and

*f*is a given function. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson (1781--1840).

**Example: **
Let us consider a classical Poisson problem

*u*(

*x, y*) has singularities of the form

*r*is the radius of a local polar coordinate centered on the corner. ■

**Example: **
■

**Example: **
■

**Example: **
Let Ω be a bounded domain in Euclidean *N*-space with a sufficiently
smooth boundary ∂Ω and let *r* denote the distance from fixed
point in Ω taken to the origin. We consider the standard Dirichlet or
Neumann problems for the Laplace operator in spherical coordinates in *N*
dimensions that depends only of the distance from the origin

*r*

^{N-1}and integration of the Poisson equation results in

*k*

_{1}is a constant of integration. For

*N*= 2, we have

*N*-dimensional ball of radius

*a*, we have for the Dirichlet porblem the equation

**Example: **
Consider the previously considered problem for the biLaplacian:

*k*

_{1}and

*k*

_{2}are constants of integrations. Clearly, the cases

*N*= 2 and

*N*= 4 must be treated separately. For

*N*= 2, we have

*N*= 4, we have

- Strang, G. and Fix, G.J., An Analysis of the Finire Element Method, Prentice-Hall, New York, 1971, Chapter 8.

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