Math 481. Vector and Tensor Analysis
The emphasis is made on rigorous presentation of concepts and definitions, focusing on examples, rather than on proofs. Topics covered include:
- Manifolds: configuration spaces, abstract differentiable manifolds, tangent spaces, tangent bundles, orientability.
- Calculus on manifolds: Vector fields, flows, tensor fields.
- Differential forms and exterior calculus.
- Singular cubes and singular chains. Integration theory on manifolds and Generalized Stokes theorem. Applications to Maxwell equations and general relativity.
- Riemannian geometry: Riemannian metrics, Riemannian connections, geodesics and (time permitting), curvature.
Prerequisites: Multivariable calculus (Math 241 or equivalent) and linear algebra (Math 415, or Math 416, or equivalent).
Main Textbook: T. Frankel, The Geometry of Physics, An Introduction, Cambridge U.P. 1997 (paperback).
Optional auxiliary texts:
- M. Spivak, Calculus on manifolds, Perseus Books, 1998 (paperback)
- W. Kühnel, Differential Geometry, Student mathematical Library, vol. 16, 2002, American Mathematical Society (paperback)
- R. Bishop and S. Goldberg, Tensor Analysis on Manifolds, Dover, 1980 (paperback)
Approved by UAC 3/14/13