Math 603-601 - Final Exam Review -- Fall 2002

Math 603-601 - Review for Final Exam

General information The exam will have six to eight questions, some with multiple parts. The test will cover the material discussed after the midterm exam; there will be no direct questions on material prior to the midterm. You will be asked to do simple derivations or verifications, and to solve problems similar to ones given for homework. You will need to know basic definitions and theorems well enough to answer such questions; you will not be asked to state any definitions or theorems. The specific topics covered are listed below. Points will be distributed roughly according to time spent on a topic. Accordingly, approximately 50% of the test will be over tensors, 30% over line and surface integrals, and 20% over PDEs. Please bring an 8½×11 bluebook.

Tensors

Curvilinear coordinates
- Generalized coordinates, cylindrical coordinates and spherical coordinates. (See §2.8 in Barisenko and Tarapov.)
- Coordinate surfaces (See Barisenko and Tarapov, §2.8.1.)
- Coordinate curves (See Barisenko and Tarapov, §2.8.2.)
Vectors and tensors in a fixed coordinate system
- The basis B
- The metric tensor
- The reciprocal basis B_r
- Vectors & components - contravariant and covariant
- Tensors & components - contravariant, covariant, and mixed
Transformation laws for vectors and tensors under changes of coordinates
- The bases B′ and B′_r
- Tensors of order 2
- Higher order tensors
The gradient operator
Physical examples of tensors
- The inertia tensor (See §2.4.3 in Borisenko and Tarapov.)
- The stress tensor (See §2.4.2 in Barisenko and Tarapov.)
- The deformation (strain) tensor (See §2.4.4 in Barisenko and Tarapov.)

Line and Surface integrals

Line integrals and Green's Theorem
Surfaces (See my notes on surfaces).
- Parametrizations of simple surfaces
- Normals - unit and standard
- Area element
- Flux and density integrals
Stokes' Theorem
- The curl in generalized coordinates and spherical coordinates
The Divergence Theorem
- The divergence in generalized coordinates and spherical coordinates

Partial differential equations.

Heat flow problems, initial value and boundary value problems (See Z/T §VI.2.)
Classification of partial differential equations (See Z/T §V.8.)
Separation of variables.
- Laplace's equation in the disk. (See Z/T § VII.2, §VII.7.)
- Vibrating string. (See Z/T §VIII.8.)
- Eigenvalue problems and eigenfunction expansions. (See Z/T §VIII.10.)