# Advanced Mathematics for Applications Mathematica Notebooks

These Mathematica notebooks are all used for teaching and illustrating concepts. Most have some "discovery learning" built in. These notebooks are used in the course Advanced Mathematics for Applications at Indiana University of Pennsylvania. This is a course that typically uses a book with a title like "Advanced Engineering Mathematics".

## Please Note: The links to the HTML versions are not working right now. The links to the notebooks are functional, however.

The order of topics of the course I teach is:
1. Vectors
Dot product, cross product, abstract vector spaces
2. Vector Differential Calculus
vector functions, curves, velocity, acceleration, curvature, vector fields, streamlines, gradient, divergence, curl
3. Vector Integral Calculus
line integrals, flows, Green's Theorem, potential theory, surface integrals, flux, Divergence Theorem, Stokes' Theorem
4. Laplace transforms
delta and Heaviside functions, DE's
5. Sturm-Liouville Theory
eigenfunction expansions, orthogonal polynomials
6. Fourier Series, Partial Differential Equations
7. Fourier Integral, Fourier Transform, Partial Differential Equations

# The Mathematica Notebooks

The notebooks are listed in the order I assign them. Right click on the download button to save.

I collected all of the functions and packages that I need for vector calculus in this one notebook. Evaluate this before each vector calculus session. This made things easy on me and my students.

This has some Mathematica basics about vectors along with some functions for plotting vectors and lists of vectors.

A notebook illustrating parametric curves, gradient, divergence and much more. This is a new and improved version of veccalc.nb.

Streamlines, equipotential curves, and a bit of complex functions.

A notebook illustrating arc length, flows, surface area, flux.

A notebook with functions to look at orthogonal coordinate systems and the div, grad, curl, in these coordinate systems. This contains Mathematica commands and functions for investigating these coordinate systems, but it is by no means self-contained.

A notebook illustrating Laplace transforms and using them to solve some differential equations.

A review of the properties of even and odd functions in anticipation of orthogonal expansions and Fourier series.

Orthogonal expansions with Legendre, Hermite, Laguerre, and Chebyshev polynomials.

Fourier series as an example of Sturm-Liouville theory. Many examples.

Fourier series via inner products. Very similar to slfourier.nb, but without the use of Sturm-Liouville theory.

Using Fourier series to solve DE's where the forcing function is expressed as a series.

Examples of some modified commands to speed up calculations of Fourier coefficients.

A notebook illustrating some basic properties of complex numbers (and the relevant Mathematica commands) that may be useful in a course in applied mathematics.

Two examples of how the complex Fourier series is equivalent its real counterpart.

Some examples of Fourier transforms and Fourier integral representation of functions.

Illustrating Laplace transform solution of PDE's.

The next seven files can be used in a linear algebra course or as a linear algebra supplement to an applied mathematics course.The spline.nb notebook is a combination of differential equations and linear algebra.

A notebook illustrating the basics of entering and manipulating matrices using Mathematica.

A Notebook illustrating properties of matrices: elementary matrices, echelon forms, inverses. Requires the package gjsf.m.

A Notebook illustrating properties of matrices: systems, subspaces, bases, Fundamental Theorem of Linear Algebra. Requires the package gjsf.m.

A Mathematica package that incorporates some nice functions: elementary row operations, elementary matrices, LU decomposition. This package is needed in linalg1.nb and linalg2.nb.