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
- I. M. Gelfand, Lectures on Linear Algebra, Dover
Publications, Inc., New York, NY. (ISBN: 0-486-66082-6)
- D. Lovelock and Hanno Rund,, Tensors, Differential Forms,
and Variational Principles, Dover Publications, Inc., New York,
NY, 1989. (ISBN: 0-486-65840-6)
- E. Zachmanoglou and D. Thoe, Introduction to Partial
Differential Equations with Applications, Dover
Publications, Inc., New York, NY. (ISBN: 0-486-65251-3)
- Other math
books from Dover.
- Syllabus
- 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
- 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
- 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
- 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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