Math 503, Fall 2006

Topics Covered in Each Lecture

 

Lecture No

Date

Content

Corresponding Reading material

1

Sep.

19

Construction of complex numbers, their addition and multiplication

-

2

 

21

Euler’s formula, polar representation and integer powers of complex numbers

Kreyszig: 602-610

3

 

26

Roots of complex numbers, definition and examples of real and complex vector spaces

Kreyszig: 610-611, 323-324
4   28 Vector subspace, span, linear independence, basis, linear operator

Kreyszig: 300, 327
5 Oct. 03 Matrix representation of linear operators in finite dimension, matrix addition and multiplication, invertible linear operators, basis transformations

-
6   05 Characterization of finite-dim vector spaces, Null space, examples: polynomial and function spaces, matrix representation of derivative operator

-
7   10 Inner product spaces, orthonormal basis, Gram-Schmidt process

-
8   12 Matrix rep. in an orthonormal basis, self-adjoint operators, eigenvalue problem -
9   17 Spectral theory for self-adjoint operators, Hamel and Schauder bases, Examples of Schauder bases, Linear diff. operators and linear ODEs, general solution of 1st order linear ODEs. Kreyszig:  59-61, 26-30
10   19 Initial value problems, existence and uniqueness theorem for solution of linear ODEs, form of the general solution of 2nd order linear ODE’s, Computing a second linearly indep. solution out of a given solution of a homogeneous 2nd order linear ODE. Kreyszig: 45-58, 73-78
11   31 Solution of non-homogeneous 2nd order linear ODE, Green's functions, sequences, series, power series, ratio test, (real) analytic functions. Power series solution of 2nd order linear ODE about an ordinary point (Statement of the theorem). Kreyszig: 78-79, 98-101, 166-175
12 Nov. 02 Hermite's eqn and Hermite polynomials as an example of the application of the method of power series solution, Frobenius method. Kreyszig: 177-187
13   09 Boundary-value problems: Dirichlet, Neumann, and Sturm-Liouville boundary conditions,  Sturm-Liouville problem, orthogonal functions, examples Kreyszig: 203-209
14   14 Review of Calculus of several variables: Directional and partial derivatives, differential, gradient and its applications and interpretations, Taylor series expansion, local minimum, maximum, and saddle points, Hessian. Kreyszig: 400-407
15   16 Review of Vector Calculus: Continuous and differentiable vector valued functions, divergence, curl, and Laplacian, coordinate transformations, line integrals, conservative forces, Green's theorem in plane. Kreyszig: 410-444
16   21 Divergence theorem in plane (with proof), Divergence theorem in R^3 (without proof), Green's identities, uniqueness theorem for Laplace's eqn. Kreyszig: 458-468
17   23 Stokes’ theorem, variation of a functional, derivation of the Euler-Lagrange eqn for a functional of a single variable. 469-473 & Riley et al: 775-777
18   28 Applications of the Euler-Lagrange eqn: Geodesics in plane, shape of a flexible cable of uniform mass density connecting two towers, Brachistochrone problem. Riley et al: 778-781
19   30 Functional derivative, Euler-Lagrange equations for n-variables, application in classical mechanics, variational solution of the eigenvalue problem Riley et al: 782, 788-796
20 Dec. 05 Partial Differential Eqns: General PDEs, Linear PDEs, Homogeneous Linear PDEs, Solution of the1st order Linear PDEs using the method of characteristics (both homogeneous and non-homogeneous cases) Strauss: 1-10
21   07 Solution of the most general 1st order Linear PDE; Second order Linear PDEs; Elliptic, hyperbolic, parabolic eqns. in 2 dimensions Strauss: 27-29
22   12  Elliptic, hyperbolic, parabolic equations in all dimensions: Vibrating string in 1+1 dimensions: Separation of variables, Fourier series solution Strauss: 29-30; Kreyszig: 535-546
23   14  D'Alambert's solution of the wave eqn in 1+1 dim., Heat equation, Laplace eqn in a rectangular region in plane Kreyszig: 548-560
24   21  Complex Fourier transform and its application in solving ODEs. Kreyszig: 506-523
25   26  Solution of the Heat eqn in R^(1+1) and wave eqn in R^{2+1) using the method of Fourier transform. Kreyszig: 562-568
26   28  Vibrating Membrane: Rectangular Membrane, Circular Membrane (Laplace's eqn in polar coordinates) Kreyszig: 569-584

Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.