Methods of Applied Mathematics II

Math 603-601 - Fall 2002

Instructor: Dr. Francis J. Narcowich
Office: 302 Milner Hall
E-mail: fnarc@math.tamu.edu
Phone: 845-7369
URL: /~francis.narcowich/
Office Hours: TR 11-12, W 12-1, and by appointment.
Catalogue Description: Math 603. Methods of Applied Mathematics II. Tensor algebra and analysis; partial differential equations and boundary value problems; Laplace and Fourier transform methods for partial differential equations. Prerequisites: MATH 601 or 311.

Time & Place
TR 9:35-10:50, ZACH 119A.
Grading
Homework, 30%
Midterm, 35%
Final Examination, 35%
Required Texts
  1. A. I. Borisenko and I. E. Tarapov, Vector and Tensor Analysis, Dover Publications, Inc., New York, NY, 1979. (ISBN: 0-486-63833-2)
  2. E. Zachmanoglou and D. Thoe, Introduction to Partial Differential Equations with Applications, Dover Publications, Inc., New York, NY. (ISBN: 0-486-65251-3)
Supplementary Texts
  1. R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Dover Publications, Inc., New York, NY, 1989. (ISBN: 0-486-66110-5)
  2. David C. Kay, Tensor Calculus. Schaum's Outline Series, McGraw-Hill, NY, 1988. (ISBN: 0-07-033484-6)
  3. Georgi P. Tolstov, Fourier Series, Dover Publications, Inc., New York, NY, 1976. (ISBN: 0-486-63317-9)

Syllabus
  1. Vector spaces
    • subspaces, examples: Rn, spaces of functions, dual spaces
    • linear independence, linear dependence, basis, dimension, coordinates, coordinate transformations; vectors and dual vectors - rank 1 tensors
    • inner products, norms, orthogonality, Gram-Schmidt orthogonalization, least-squares approximation, QR factorization
    • basic ideas of approximation in function spaces - convergence, sequences, series of orthogonal functions
      • Fourier series
      • Legendre polynomials, other orthogonal functions
  2. Linear transformations, matrices, and tensors
    • linear transformations - matrix representation
    • spectral theory - eigenvalues, eigenvectors, diagonalization, self-adjoint matrices, singular value decomposition, Jordan normal form; applications to linear systems of ODEs
    • tensors - invariance under coordinate transformations; multi-linear functions; contravariant, covariant, and mixed-type tensors; tensor algebra
  3. Vector and tensor analysis
    • limits, continuity, gradient, divergence, curl - coordinate independent definitions; scalar, vector, and tensor valued functions
    • integral calculus - Green's theorem, divergence theorem, Stokes' theorem
    • div, grad, curl in general curvilinear coordinates
    • invariant equations in fluid dynamics
  4. Partial differential equations
    • PDEs arising applications - wave, heat, and potential equations
    • D'Alembert's solution to the wave equation
    • separation of variables - eigenvalue problems, Sturm-Liouville problems, orthogonal expansions, Fourier and Laplace transforms
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