Course Materials - Mathematics Topics

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Where possible, links to materials are divided into both Physics Topics and Mathematics Topics. Additional class notes are planned. Links to other course notes are under the Links page.

Mathematics Topics

Mathematics Background

• Integrals you should know

• SOS Math - Calculus

• Summary Sheet on Exponential and Logarithms

• Summary of Basic Derivatives and Integrals

• Integration by Substitution

• Integration by parts - SOS Math, Examples, Visual Calculus

• Improper Integrals - SOS Math, Examples + Applets

• The Wolfram Integrator

• Trigonometry

  • SOSMath on Trigonometry

  • Trigonometric Functions of Common Angles

  • Trigonometry: A Day at the Track

  • Derivatives of the Trigonometric Functions

  • Visual Math - Derivatives

  • Trigonometric Formulae

• Hyperbolic Functions

  • Wikipedia

  • MathWorld

  • Identities and here

  • Inverse hyperbolic functions

Ordinary Differential Equations

• History Lecture

• Separable Equations

  • More on Separable Equations 1, 2, 3

• Exact Differential Equations

• First Order Linear

  • More - 1, 2, 3

• Applications

• A Fish Story

• Second Order Differential Equations

• Analytical Methods for Second Order Linear Ordinary Differential ... and Examples

• More Second Order ODEs

• Variation of Parameters

• Theoretical Description of Spacecraft Flybys by Variation of ...

• Examples

  • Undetermined Coefficients

  • Variation of Parameters

Numerical Solutions

• Euler's Method Applet, Another One

• Matlab in DEs

• Maple File Euler.mws

Matrices and Systems of Differential Equations

• Romance Maple File - Class Demo of Direction Fields

• Linear Systems - SOS Math

• An Introduction to Matrices

• Matrices

• Eigenvalues and Eigenvectors - SOS Math

• Qualitative Analysis - SOS Math

• Phase Portraits - Applet or try a Mathlet

• Models of Species Interaction

• Interactive Differential Equations

• Solving Linear Systems - Summary of Matrix Methods - R. Herman

Cauchy-Euler Equations

• SOS Math

• Recipe - PDF - Order 2, PDF File 2, Problems and Solutions!

Fourier Series

• Prelude for Strings - The Origin of Spectral Theory

• Dr. Herman's MAT 367 Site

• Fourier Series - Eric Weinstein's World of Math

• SOS Math - Fourier Series

• Listen to Fourier Series

• Fourier Series and Convergence

• Fourier Series and Waves

• Maple Worksheet - fseries.mws, EvenOdd.mws, PlotFS.mws

• Gibbs Phenomenon - SOS Math

• Fourier Series Applet

Separation of Variables

• Solution of the Heat Equation

• Heat Equation Module

• Special Trigonometric Bases - Harvey Mudd College

• Products of Trigonometric Functions, also at ThinkQuest

• Lectures on Laplace's Equation at Herod's Asynchronous site

  •  Laplace's Equation on a Rectangle

  • Insulated Boundaries

  •  The Structure of Solutions

  • The Laplacian in Polar Coordinates

  • On a Ring and On a Half-Disk

• More on Laplace's Equation by Alan S. Jones part of PDE site

Complex Numbers and Functions

• Online text: Complex Analysis by George Cain

• Free Textbook on Complex Analysis

• Complex Numbers

• Short Course on Complex Numbers

• An Introduction to Complex Numbers

• Irrotational Flow around objects

• Conformal Mapping Example

• Plotting Functions of a Complex Variable

• Ideal Flow Machine

• Visual Complex Analysis, The Book

• Zipper

• Conformal Mapping and Minimal Surfaces

• Software Collection for Complex Variables

• Visualizing Complex Functions Using Domain Coloring

• Graphics for Complex Analysis

• Visualizing Complex Functions

• Complex Function Viewer - Java

• The Chaos Hypertextbook

Complex Integration

• Online text: Complex Analysis by George Cain

• Free Textbook on Complex Analysis

Fourier Transform

• Dirac Delta Function - Mathworld, K. Bryan's The Delta Function

• Fourier Transform - Eric Weinstein's World of Math

• The Fourier Integral

• Fourier Transforms

• Maple File for Fourier Transforms

• Solving Laplace's Equation Using Fourier Transforms

• Examples - Digital Geophysical Analysis

• Fourier Transforms for Image Analysis

Laplace Transform

• Laplace Transform Table - MS Word

• Practice Laplace Transform Problems

• Introduction to Laplace Transforms for Engineers

• Mathworld

• Maple Example - ODE

• Laplace Transforms and Initial Value Problems

• IVP - Animation

• Tables

  • Collections of Worksheets! http://www.equationsheet.com/ 

  • http://www.vibrationdata.com/Laplace.htm 

  • eFunda: Laplace Transform Table

  • http://cnyack.homestead.com/files/alaplace/laptre.htm 

  • Table of Laplace Transform Properties 

  • http://www.sosmath.com/tables/laplace/laplace.html 

• Examples

• Laplace Transforms and Circuit Analysis

• Application to Magnetically Coupled Circuits

• Laplace Transform for Summing Series

• Application to a System of ODEs

• Poles of Laplace Transform and Behavior of Solutions

• Poles in Matlab

Vector Calculus

• Vector Review

• Wave Equations

• Div, Grad, Curl and Maxwell's Equations - PowerPoint

• Electromagnetic Waves - The Physics Hypertextbook

• Vector Identities or Here

Sturm-Liouville Problems

• Sturm-Liouville Eigenvalue Problem
• The Sturm Liouville Problem

Curvilinear Coordinates

• Curvilinear Coordinates
• Orthogonal Coordinate Systems
  1. Cartesian coordinates,
  2. spherical coordinates,
  3. cylindrical coordinates,
  4. elliptic cylindrical coordinates,
  5. parabolic cylindrical coordinates,
  6. bipolar cylindrical coordinates
  7. prolate spheroidal coordinates,
  8. oblate spheroidal coordinates,
  9. parabolic coordinates,
  10. toroidal coordinates.
  11. bispherical coordinates
  12. confocal ellipsoidal coordinates,
  13. conical coordinates,
  14. confocal paraboloidal coordinates,

Bessel Functions and Vibrating Membranes

• Rectangular Membrane, Circular Membrane, Annular Membrane
• Circular Membrane Modes

• Vibrations and Waves - Animations

• Maple Files annulus, vibdrum

• Essentials of Bessel Functions

• Bessel Functions and a Vibrating Drum 

• Plots of Bessel Functions

Spherical Harmonics

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