Table of Contents

Introduction and Review
What Do I Need To Know From Calculus?
What I Need From My Intro Physics Class?
Technology and Tables
Appendix: Dimensional Analysis
Problems

Free Fall and Harmonic Oscillators
Free Fall
First Order Differential Equations
The Simple Harmonic Oscillator
Second Order Linear Differential Equations
LRC Circuits
Damped Oscillations
Forced Systems
Cauchy-Euler Equations
Numerical Solutions of ODEs
Numerical Applications
Linear Systems
Problems

Linear Algebra
Finite Dimensional Vector Spaces
Linear Transformations
Eigenvalue Problems
Matrix Formulation of Planar Systems
Applications
Appendix: Diagonalization and Linear Systems
Problems

Nonlinear Dynamics
Introduction
The Logistic Equation
Autonomous First Order Equations
Bifurcations for First Order Equations
Nonlinear Pendulum
The Stability of Fixed Points in Nonlinear Systems
Nonlinear Population Models
Limit Cycles
Nonautonomous Nonlinear Systems
Exact Solutions Using Elliptic Functions
Problems

The Harmonics of Vibrating Strings
Harmonics and Vibrations
Boundary Value Problems
Partial Differential Equations
The 1D Heat Equation
The 1D Wave Equation
Introduction to Fourier Series
Fourier Trigonometric Series
Fourier Series Over Other Intervals
Sine and Cosine Series
Solution of the Heat Equation
Finite Length Strings
The Gibbs Phenomenon
Green’s Functions for 1D Partial Differential Equations
Derivation of Generic 1D Equations
Problems

Non-sinusoidal Harmonics and Special Functions
Function Spaces
Classical Orthogonal Polynomials
Fourier-Legendre Series
Gamma Function
Fourier-Bessel Series
Sturm-Liouville Eigenvalue Problems
Nonhomogeneous Boundary Value Problems - Green’s Functions
Appendix: The Least Squares Approximation
Appendix: The Fredholm Alternative Theorem
Problems

Complex Representations of Functions
Complex Representations of Waves
Complex Numbers
Complex Valued Functions
Complex Differentiation
Complex Integration
Problems

Transform Techniques in Physics
Introduction
Complex Exponential Fourier Series
Exponential Fourier Transform
The Dirac Delta Function
Properties of the Fourier Transform
The Convolution Operation
The Laplace Transform
Applications of Laplace Transforms
The Convolution Theorem
The Inverse Laplace Transform
Transforms and Partial Differential Equations
Problems

Vector Analysis and EM Waves
Vector Analysis
Electromagnetic Waves
Curvilinear Coordinates
Tensors
Problems

Extrema and Variational Calculus
Stationary and Extreme Values of Functions
The Calculus of Variations
Hamilton’s Principle
Geodesics
Problems

Problems in Higher Dimensions
Vibrations of Rectangular Membranes
Vibrations of a Kettle Drum
Laplace’s Equation in 2D
Three Dimensional Cake Baking
Laplace’s Equation and Spherical Symmetry
Schrödinger Equation in Spherical Coordinates
Solution of the 3D Poisson Equation
Green’s Functions for Partial Differential Equations
Problems

Review of Sequences and Infinite Series
Sequences of Real Numbers
Convergence of Sequences
Limit Theorems
Infinite Series
Convergence Tests
Sequences of Functions
Infinite Series of Functions
Special Series Expansions
The Order of Sequences and Functions
Problems