Announcements

Lecture No	Date	Content	Corresponding Reading material
1	Sep. 18	Construction of complex numbers, their addition and multiplication
2	Sep. 20	Euler’s formula, polar representation, integer powers and roots of complex numbers	Kreyszig: 602-612
3	Sep. 25	Logarithm and complex powers of a complex number, trigonometric and hyperbolic functions of a complex variable and their inverse; Review of Linear Algebra: Real and complex vector spaces, subspaces, span, linear independence, basis, dimension	Kreyszig: 623-633, 323-329
4	Sep. 27	A basis for the vector space of polynomials, linear operators, derivative operator, matrix representation of a linear operator mapping finite-dimensional vector spaces, zero and identity operator, vector space of linear operators	-
5	Oct. 02	Invertible linear operators, characterization of finite-dim vector spaces, null space, example: derivative and differential operators, C^n(R) spaces, solution space of homogeneous linear ODEs as null spaces; Inner product spaces: inner product, its associated norm and metric	-
6	Oct. 04	Orthonormal bases, Gram-Schmidt process, Matrix rep. in an orthonormal basis, matrix representation of the derivative operator acting on a finite span of sin's and cos's, Hermitian matrices	-
7	Oct. 09	Isometries and unitary operators, characterization of finite-dimensional inner product spaces, self-adjoint operators, eigenvalue problem, spectral theory for self-adjoint operators, convergence of sequences and series in an inner product space, dense subsets, approximating analytic and continuous functions by polynomials	-
8	Oct. 16	Hamel and Schauder bases, Examples of Schauder bases, Linear differential operators and linear ODEs, general solution of 1st order linear ODEs, Initial value problems, existence and uniqueness theorem for the solution of linear ODEs, form of the general solution of 2nd order linear ODE’s, Computing a second linearly independent solution out of a given nonzero solution for a homogeneous 2nd order linear ODE	Kreyszig: 26-30, 45-52, 73-78
9	Oct. 18	Solving second order linear homogeneous ODEs with constant coefficients (using the factorization method); Solving non-homogeneous 2nd order linear ODE (using variation of parameters), Green's function	Kreyszig: 53-61, 78-79, 98-104.
10	Oct. 23	Power series and their convergence, ratio test, (real) analytic functions. Power series solution of 2nd order linear ODE about an ordinary point, Hermite's eqn and Hermite polynomials	Kreyszig: 166-182
11	Oct. 25	Frobenius method, an application	Kreyszig: 182-203
12	Nov. 01	Boundary-value problems: Dirichlet, Neumann, and Sturm-Liouville boundary conditions, Sturm-Liouville problem, orthogonal functions, examples	Kreyszig: 203-209
13	Nov. 06	Review of Calculus of several variables: Directional and partial derivatives, differential, gradient and its applications and interpretations, Taylor series expansion, linear and quadratic approximations, Hessian, stationary points, local minimum, maximum, and saddle points	Kreyszig: 400-407 and 937-938
14	Nov. 08	Eigenvalues of the Hessian and the local minimum, maximum, and saddle points. Review of Vector Calculus: Continuous and differentiable vector-valued functions, divergence, curl, and Laplacian, coordinate transformations, line integrals, conservative forces	Kreyszig: 407-428
15	Nov. 13	Exact differentials and conservative forces, Kronecker delta- and Levi Civita epsilon-symbols and their use in deriving various identities of vector calculus, Green's theorem in plane	Kreyszig: 439-442
16	Nov. 15	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	Nov. 20	A characterization theorem for exact differentials in two-dimensions, Stokes’ theorem (without proof) and its verification using a specific example, Functionals: motivation, examples	Kreyszig: 429-432 and 468-473
18	Nov. 22	Variation of a functional, derivation of the Euler-Lagrange eqn for a functional of a single variable, applications: geodesics in plane, shape of a flexible cable of uniform mass density connecting two towers	Riley et al: 775-781
19	Nov. 29	Functional derivative of a functional, variational solution of the eigenvalue problem.	Riley et al: 790-796
20	Dec. 04	Variation of a functional dependent on n-variables; 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), examples	Strauss: 1-10
21	Dec. 06	Second order Linear PDEs; Elliptic, hyperbolic, parabolic eqns.; Vibrating string in 1+1 dimensions: Separation of variables	Strauss: 29-30; Kreyszig: 535-543
22	Dec. 11	Fourier series solution of the wave equation for the Vibrating string in 1+1 dimensions	Kreyszig: 543-546
23	Dec. 13	D'Alambert's solution of the wave eqn in 1+1 dim., Heat equation, Laplace equation in a rectangular region in plane; Fourier transform and its existence	Kreyszig: 548-560 and 518-520
24	Dec. 18	Application of Fourier transform in solving ODEs, Inverse Fourier transform, solution of the Heat eqn in R^(1+1) using the method of Fourier transform	Kreyszig: 520-522 and 562-568
25	Dec. 27	Solution of wave eqn in R^{2+1) using the method of (double) Fourier transform; Vibrating Rectangular Membrane and double Fourir series	Kreyszig: 569-578

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