Syllabus

The course is designed to give you some experience in the areas of mathematics that may be important to you at various times in your career. This course differs from others in that you may have no interest in a particular technique at this point in your career, and yet years from now your familiarity with it may allow you to explore a new field of astrophysics, or a new aspect of an old one.

The topics I have picked are subject to revision as we see how things go during the semester. They are based on a survey of the departmental and observatory faculty, research staff, and graduate students. They represent a sort of consensus of the most useful topics for astrophysics.

We each bring very diverse interests and experiences to this course, as well as vastly different mathematical preparation. If you are among the better prepared, this course is a way to fill-in the holes you have in your mathematical preparation, and review and refine your understanding of the areas you are already familiar with. If you have not been previously exposed to most of this material, you may have to spend a little more time reading some of the recommended texts. If this is you, be consoled that there is no better time in your career to do this than now.

Mathematics is a tool for doing science. It is to science what technique, or "chops", are to musicians. This is a chance to improve your chops. Be careful though, because the analogy works in another way as well. Your chops are not static. If you don't practice them, you will lose them. I'm living proof of that! So in the future, you must revisit these subjects occasionally to keep your understanding sharp and fresh. Make this easy on yourself, and keep your notes.

We will divide into groups, based on research interests. We will select applications from each area of research represented. Your group will make a short presentation of these applications. Your grade will be the average of these scores (weighted 60%) plus your class participation score (weighted 40%). Read the notes in advance of each day's lecture and bring your questions. I will then cover the key elements from the notes and respond to your questions. At the beginning of each lecture a student, selected by me, will summarize the previous lecture. We will discuss the possibility of exams and problem sets in class.

**Vector Analysis**

- A Brief review of Vector Analysis: Gradient, Divergence Curl, and Integrations

- Some Useful Theorems: Gauss', Stokes', and Helmholtz's

- A Brief review of Vector Analysis: Gradient, Divergence Curl, and Integrations
**Vector Spaces and Matrices**

- Linear Vector Spaces

- Linear Operators

- Introduction to Matrices

- Coordinate Transformations

- Eigenvalue Problems

- Diagonalization of Matrices

- Spaces of Infinite Dimensionality, Hilbert Spaces

- Linear Vector Spaces
**An Introduction to Tensor Analysis and Differential Geometry**

- Cartesian Tensors in Three-Space

- Coordinate Transformations and General Tensor Analysis

- The Metric Tensor

- Geodesics

- Christoffel Symbols

- Covariant Derivatives

- Parallel Transport

- Geodesics Through Parallel Transport

- The Riemann-Christoffel Curvature Tensor

- Parallel Transport around a Closed Loop and Curvature

- The Absolute Derivative, Geodesic Deviation and Curvature

- Cartesian Tensors in Three-Space
**Calculus of Variations**

- Euler-Lagrange Equation

- Generalizations of the Basic Problem

- Euler-Lagrange Equation
**Infinite Series**

- Fundamental Concepts

- Convergence Tests

- Familiar Series

- Taylor's Expansion

- Transformation of Series

- Fundamental Concepts
**Complex Analysis Part I: Analytic Functions**

- Complex Algebra

- Cauchy-Riemann Conditions

- Cauchy's Integral Theorem and Formula

- Laurent Expansions

- Complex Algebra

**Complex Analysis Part II: Calculus of Residues**

- Singularities

- Calculus of Residues

- The Evaluation of Real Integrals

- Singularities
**Probability and Statistics**

- Introduction

- Fundamental Probability Laws

- Combinations and Permutations

- The Binomial, Poisson, and Gaussian Distributions

- General Properties of Distributions

- Fitting of Exprimental Data

- Introduction
**Eigenfunctions, Eigenvalues, and Green's Functions**

- Simple Examples of Eigenvalue Problems

- General Discussion

- Solutions of Boundary-Value Problems as Eigenfunction Expansions

- Inhomogeneous Problems, Green's Functions

- Green's Functions in Electrodynamics

- Simple Examples of Eigenvalue Problems
**Evaluations of Integrals**

- Elementary Methods

- Use of Symmetry Arguments

- Contour Integration

- Tabulated Integrals

- Approximate Expansions

- Saddle-Point Methods

- Elementary Methods
**Integral Transforms**

- Fourier Series

- Fourier Transforms

- Laplace Transforms

- Other Transform Pairs

- Applications of Integral Transforms

- Fourier Series
**Perturbation Theory**

- Conventional Nondegenerate Theory

- A Rearranged Series

- Degenerate Perturbation Theory

- Conventional Nondegenerate Theory
**Special Functions**

- Legendre Functions

- Bessel Functions

- Hypergeometric Functions

- Confluent Hypergeometric Functions

- Mathieu Functions

- Elliptic Functions

- Legendre Functions