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
- A. I. Borisenko and I. E. Tarapov, Vector and Tensor
Analysis, Dover Publications, Inc., New York, NY, 1979. (ISBN:
0-486-63833-2)
- 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
- R. Aris, Vectors, Tensors, and the Basic Equations of Fluid
Mechanics, Dover Publications, Inc., New York, NY, 1989. (ISBN:
0-486-66110-5)
- David C. Kay, Tensor Calculus. Schaum's Outline
Series, McGraw-Hill, NY, 1988. (ISBN: 0-07-033484-6)
- Georgi P. Tolstov, Fourier Series, Dover
Publications, Inc., New York, NY, 1976. (ISBN: 0-486-63317-9)
- Syllabus
- 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
- 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
- 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
- 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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