
Syllabus
Approach
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 fillin
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.
Grading
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.
Current Detailed Contents (Subject to Revision)
 Vector Analysis
 A Brief review of Vector Analysis: Gradient, Divergence Curl, and Integrations
 Some Useful Theorems: Gauss', Stokes', and Helmholtz's
 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
 An Introduction to Tensor Analysis and Differential Geometry
 Cartesian Tensors in ThreeSpace
 Coordinate Transformations and General Tensor Analysis
 The Metric Tensor
 Geodesics
 Christoffel Symbols
 Covariant Derivatives
 Parallel Transport
 Geodesics Through Parallel Transport
 The RiemannChristoffel Curvature Tensor
 Parallel Transport around a Closed Loop and Curvature
 The Absolute Derivative, Geodesic Deviation and Curvature
 Calculus of Variations
 EulerLagrange Equation
 Generalizations of the Basic Problem
 Infinite Series
 Fundamental Concepts
 Convergence Tests
 Familiar Series
 Taylor's Expansion
 Transformation of Series
 Complex Analysis Part I: Analytic Functions
 Complex Algebra
 CauchyRiemann Conditions
 Cauchy's Integral Theorem and Formula
 Laurent Expansions
 Complex Analysis Part II: Calculus of Residues
 Singularities
 Calculus of Residues
 The Evaluation of Real Integrals
 Probability and Statistics
 Introduction
 Fundamental Probability Laws
 Combinations and Permutations
 The Binomial, Poisson, and Gaussian Distributions
 General Properties of Distributions
 Fitting of Exprimental Data
 Eigenfunctions, Eigenvalues, and Green's Functions
 Simple Examples of Eigenvalue Problems
 General Discussion
 Solutions of BoundaryValue Problems as Eigenfunction Expansions
 Inhomogeneous Problems, Green's Functions
 Green's Functions in Electrodynamics
 Evaluations of Integrals
 Elementary Methods
 Use of Symmetry Arguments
 Contour Integration
 Tabulated Integrals
 Approximate Expansions
 SaddlePoint Methods
 Integral Transforms
 Fourier Series
 Fourier Transforms
 Laplace Transforms
 Other Transform Pairs
 Applications of Integral Transforms
 Perturbation Theory
 Conventional Nondegenerate Theory
 A Rearranged Series
 Degenerate Perturbation Theory
 Special Functions
 Legendre Functions
 Bessel Functions
 Hypergeometric Functions
 Confluent Hypergeometric Functions
 Mathieu Functions
 Elliptic Functions
