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MATH 481 Intro to Differential Geometry (Vector and Tensor Analysis)

The basic tools of differential geometry will be introduced at the undergraduate level, by focusing on examples. This is a good first course for those interested in, or curious about, modern differential geometry, and in applying differential geometric methods to other areas.

Manifolds: configuration spaces, differentiable manifolds, tangent spaces, tangent bundles, orientability.

Calculus on manifolds: Vector fields, flows, tensor fields.

Differential forms and exterior calculus.

Integration theory: Generalized Stokes theorem, de Rham cohomology.

Riemannian geometry: Riemannian metrics, geodesics.

Text: The Geometry of Physics, An Introduction, T. Frankel, Cambridge U.P. 1997

Or

CS 457 Numerical Methods II

Orthogonalization methods for linear least squares problems. QR factorization and singular value decomposition

Iterative methods for systems of linear algebraic equations. Stationary iterative methods. Krylov subspace methods

Eigenvalue problems. Power, inverse power, and QR iterations. Krylov subspace methods

Nonlinear equations and optimization in n dimensions. Newton and Quasi-Newton methods. Nonlinear least squares

Initial and boundary value problems for ordinary differential equations. Accuracy and stability. Multistep methods for initial value problems. Shooting, finite difference, collocation, and Galerkin methods for boundary value problems

Partial differential equations. Finite difference methods for heat, wave, and Poisson equations. Consistency, stability, and convergence

Fast Fourier transform. Trigonometric interpolation. Discrete Fourier transform. FFT algorithm