Methods of Applied Mathematics I

Math 601-604 - Fall 2000

Instructor: Dr. Francis J. Narcowich
Office: 302 Milner Hall
E-mail: fnarc@math.tamu.edu
Phone: 845-7369
URL: /~francis.narcowich/
Office Hours: TR 1-2 pm, W 2-3 pm, and by appointment.
Catalogue Description: Vector spaces, matrices, tensors, vector & tensor analysis, PDEs.
PREREQUISITE: MATH 308 and MATH 311. (MEEN Section)
Time & Place
TR 11:10-12:25, ENPH 214.
Grading
Homework, 30%
Midterm, 35% (Thursday, October 19)
Final Examination, 35% (Friday, December 8, 3-5 pm)
Texts
  1. I. M. Gelfand, Lectures on Linear Algebra, Dover Publications, Inc., New York, NY. (ISBN: 0-486-66082-6)
  2. D. Lovelock and Hanno Rund,, Tensors, Differential Forms, and Variational Principles, Dover Publications, Inc., New York, NY, 1989. (ISBN: 0-486-65840-6)
  3. E. Zachmanoglou and D. Thoe, Introduction to Partial Differential Equations with Applications, Dover Publications, Inc., New York, NY. (ISBN: 0-486-65251-3)
  4. Other math books from Dover.

Syllabus
  1. Vector spaces. (3 weeks.)
    • subspaces, examples: Rn, spaces of functions, dual spaces
    • linear independence, linear dependence, basis, dimension
    • inner products, Schwarz's inequality, norms, Gram-Schmidt orthogonalization
    • basic ideas of approximation in function spaces - convergence, sequences, Cauchy sequences, compactness
  2. Matrices and tensors. (3 weeks.)
    • linear transformations - matrix representation
    • spectral theory - eigenvalues, eigenvectors, Cayley-Hamilton theorem, diagonalization, self-adjoint matrices, Jordan normal form
    • decompositions - LU, QR, SVD; application to least squares
    • multilinear functions - tensor products; contravariant, covariant, and mixed-type tensors
  3. Vector and tensor analysis. (5 weeks.)
    • scalar, vector, and tensor valued functions
    • limits, continuity, gradient, divergence, curl - coordinate independent definitions
    • integral calculus - Green's theorem, divergence theorem, Stokes' theorem
    • Frechet derivative - Euler equations and calculus of variations
    • orthogonal curvilinear coordinates
  4. Partial differential equations. (3 weeks.)
    • PDEs arising applied mechanics - wave, heat, and potential eqautions
    • D'Alembert's solution to the wave eqaution
    • separation of variables - eigenvalue problems, Sturm-Liouville problems, orthogonal expansions
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