Syllabus Math 481


Math 481. Vector and Tensor Analysis
Instructor Syllabus

The emphasis is made on rigorous presentation of concepts and definitions, focusing on examples, rather than on proofs. Topics covered include:

  1. Manifolds: configuration spaces, abstract differentiable manifolds, tangent spaces, tangent bundles, orientability.
  2. Calculus on manifolds: Vector fields, flows, tensor fields.
  3. Differential forms and exterior calculus.
  4. Singular cubes and singular chains. Integration theory on manifolds and Generalized Stokes theorem. Applications to Maxwell equations and general relativity.
  5. 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:

  1. M. Spivak, Calculus on manifolds, Perseus Books, 1998 (paperback)
  2. W. Kühnel, Differential Geometry, Student mathematical Library, vol. 16, 2002, American Mathematical Society (paperback)
  3. R. Bishop and S. Goldberg, Tensor Analysis on Manifolds, Dover, 1980 (paperback)


Approved by UAC 3/14/13