Solutions of the exam paper, with comments.
Vectors and tensors. Differential operators in general coordinates. (4.5 lectures.)
Besides the books mentioned below, materials for lectures on this section borrowed heavily from S. weinberg, Gravitation and Cosmology, Sec.4.
Vector space. Metric space. Function space. (4.5 lectures.)
Besides Joglekar, the lectures also borrowed from K. Hoffman & R. Kunze, Linear Algebra, Secs. 2 and 8, and in particular from the discussion on function space in P. Dennery & A. Krzywicki, Mathematics for Physicists, Ch. III. Here's the mathematica notebook we used in the class for basis transformations.
Linear algebra. Eigenvalue problem. (5 lectures.)The references for the lectures are the same as those for vector space. Also G. Strang, Linear Algebra and its Applications, has some nice examples.
Introduction to group theory. (3 lectures.)
For the lectures I used notes of an old lecture course by Sunil Mukhi. The material is standard, and can be obtained from any group theory book, certainly from M. Hamermesh, Group theory and its application to physical problems, or from Wu-Ki Tung, Group Theory in Physics.
Complex analysis. Calculus of residues. (8 lectures.)
Besides Joglekar, Arfken and Dennery & Krzywicki, I also used S. Lang, Complex Analysis, for the lectures.
Linear differential equations. Partial differential equations. Green's function.
| Lect. 1 | 03.08 | Coordinate transformation and vectors. Derivative operators. Gauss' and Stokes' theorems. | |
| Lect. 2 | 08.08 | Cartesian tensors. Tensor product and contraction. Lorentz transformation. | |
| Lect. 3 | 10.08 | Covariant and contravariant vectors, Lorentz tensors. | |
| Lect. 4 | 17.08 | General coordinate transformations, tensor algebra. Covariant derivative. Gradient, divergence, curl. | |
| Lect. 5 | 24.08 | Differential operators in orthogonal coordinates. Vector space. | First assignment. |
| Lect. 6 | 05.09 | Subspace. Linear independence of vectors and basis. Dimension of vector space. Inner product, metric; Hilbert space. | |
| Lect. 7 | 07.09 | Orthogonality; orthonormal basis and Gram-Schmidt process. Dual space. | |
| Lect. 8 | 12.09 | Function space; Lebesgue integral; L2 as Hilbert space. Orthonormal basis. | |
| Lect. 9 | 14.09 | Orthogonal Polynomials. Fourier series. | Second assignment. |
| Lect. 10 | 19.09 | Linear transformation in vector space and matrices. Range and nullity. LU decomposition. Row-reduced echelon form. | |
| Lect. 11 | 21.09 | Inverse and Gauss-Jordan method. Determinant. Linear functional and dual space. Adjoint. Hermitian and unitary operators. | |
| Lect. 12 | 26.09 | Eigenvalues and Eigenvectors. Secular equation. | |
| Lect. 13 | 28.09 | Similarity transformation. Diagonalization. Unitary transformation. Eigenbasis of normal matrix. | |
| Lect. 14 | 29.09 | Cayley-Hamilton theorem. Generalized eigenvectors and Jordan canonical form. | |
| Lect. 15 | 10.10 | Groups. Multiplication table. Subgroup, cosets, quotient group. | |
| Lect. 16 | 12.10 | Reducible and irreducible representation. Unitary representation. Schur's Lemma. | Third assignment. |
| Lect. 17 | 19.10 | Orthogonality relations. Character; character table. Irrep and degeneracy of energy eigenstates. | |
| Lect. 18 | 24.10 | Functions of complex variables. Limit, differentiability, Cauchy-Riemann conditions. | |
| Lect. 19 | 31.10 | Complex series and convergence. Power series. Differentiability of power series. Complex integration. Cauchy's theorem and Integral formula. | |
| Lect. 20 | 02.11 | Gourset's proof of Cauchy's theorem. Taylor series of differentiable functions. Laurent series. | |
| Lect. 21 | 03.11 | Isolated singularities. Residue and residue theorem. Real integrals. | |
| Lect. 22 | 09.11 | Fourier transform. Principal value integral. Series sum using contour. | Fourth assignment. |
| Lect. 23 | 14.11 | Multivalued functions, branch cut and Riemann surface. Analytic continuation. | |
| Lect. 24 | 16.11 | Conformal transformations. Method of steepest descent. | |
| Lect. 25 | 21.11 | Examples: steepest descent method and use of conformal transformations. |
Mathematical Physics, by S. Joglekar. Very lucid book in two volumes. Mathematical Methods for Physicists, by Arfken and Weber. A book with lots of physical problems.
Both of these books cover most of our course material. For specific topics I will follow other books, and give the references as they come. But if you know most of the material in one of these two books, you should drop the course.