Physics 464/511, Fall 2019

Mathematical Physics

Lecture Notes

Anirvan Sengupta

Lecture Notes

This is a sketch of the materials to be covered. The lecture notes will be posted after my lectures. I indicate chapters and sections of Vaughn that roughly correspond to the material. Some material in the notes have not been presented in the class.

Lecture Dates Topic Notes Vaughn Chapters and Sections
1 Sep. 4 Preliminaries, real and complex numbers, sequences Notes 1.1

2 Sep. 6, 19 Sequences, series, monotonic sequences, limsup and liminf Notes 1.1-1.2

3 Sep. 11 Cauchy sequences, open sets, closed sets, limit points Notes 1.1-1.2

4 Sep. 13 Tests of convergence, infinite products Notes 1.2

5 Sept. 18 Sequences and series of functions Notes 1.3-1.4

6 Sept. 20 Vector space introduction, bases, orthonomality Notes 2.1.1-2.1.2

7 Sept. 27 Orthonormal system continued, vector space sums, sequences Notes 2.1.2-2.1.4

8 Oct. 2 Operators, basis change, diagonalization, eigenvalue problem, normal operators Notes 2.2.3-2.3.3

9 Oct. 4, 9 Topology, manifolds Notes 3.1.1

10 Oct. 11 Examples of manifolds, tangent space Notes 3.1.2-3.2.2

11 Oct. 16, 18 Cotangent space, 1-forms and tensors. Notes 3.2.3-3.2.5

12 Oct. 18, 23 Integral curves, Lie derivatives Notes 3.2.5-3.2.6

13 Oct. 23, 25 Wedge product, exterior algebra, integration on manifolds Notes 3.3.1

14 Oct. 30 Exterior derivative, Stokes theorem Notes 3.3.2-3.3.4

15 Nov. 1 Metric tensor, Laplacian, geodesics Notes 3.4.1-3.4.6

16 Nov. 6 Analytic functions, Cauchy-Riemann conditions, Cauchy theorem Notes 4.1.1-4.2.2

17 Nov. 8 Cauchy integral formula, Taylor series, global properties Notes 4.2.3-4.3.3

18 Nov. 13, 15 Laurent series, calculus of residues, application to real integrals Notes 4.3.4-4.4.2, A.1-A.2
19 Nov. 20, 22 Differential equations in the complex plane Notes 5.1-5.3
20 Nov. 27 Frobenius method, Legendre's equation, Bessel's equation Notes 5.4-5.6
21 Dec. 4 Hilbert spaces Notes 6.1
22 Dec. 4, 6 Measure theory, Fourier expansion Notes 6.2-6.4

28 Dec. 6 Operators on Hilbert spaces, partial differential equations Notes 7.1-7.2, 8.1-8.3

Lecture	Dates	Topic	Notes	Vaughn Chapters and Sections
1	Sep. 4	Preliminaries, real and complex numbers, sequences	Notes	1.1
2	Sep. 6, 19	Sequences, series, monotonic sequences, limsup and liminf	Notes	1.1-1.2
3	Sep. 11	Cauchy sequences, open sets, closed sets, limit points	Notes	1.1-1.2
4	Sep. 13	Tests of convergence, infinite products	Notes	1.2
5	Sept. 18	Sequences and series of functions	Notes	1.3-1.4
6	Sept. 20	Vector space introduction, bases, orthonomality	Notes	2.1.1-2.1.2
7	Sept. 27	Orthonormal system continued, vector space sums, sequences	Notes	2.1.2-2.1.4
8	Oct. 2	Operators, basis change, diagonalization, eigenvalue problem, normal operators	Notes	2.2.3-2.3.3
9	Oct. 4, 9	Topology, manifolds	Notes	3.1.1
10	Oct. 11	Examples of manifolds, tangent space	Notes	3.1.2-3.2.2
11	Oct. 16, 18	Cotangent space, 1-forms and tensors.	Notes	3.2.3-3.2.5
12	Oct. 18, 23	Integral curves, Lie derivatives	Notes	3.2.5-3.2.6
13	Oct. 23, 25	Wedge product, exterior algebra, integration on manifolds	Notes	3.3.1
14	Oct. 30	Exterior derivative, Stokes theorem	Notes	3.3.2-3.3.4
15	Nov. 1	Metric tensor, Laplacian, geodesics	Notes	3.4.1-3.4.6
16	Nov. 6	Analytic functions, Cauchy-Riemann conditions, Cauchy theorem	Notes	4.1.1-4.2.2
17	Nov. 8	Cauchy integral formula, Taylor series, global properties	Notes	4.2.3-4.3.3
18	Nov. 13, 15	Laurent series, calculus of residues, application to real integrals	Notes	4.3.4-4.4.2, A.1-A.2
19	Nov. 20, 22	Differential equations in the complex plane	Notes	5.1-5.3
20	Nov. 27	Frobenius method, Legendre's equation, Bessel's equation	Notes	5.4-5.6
21	Dec. 4	Hilbert spaces	Notes	6.1
22	Dec. 4, 6	Measure theory, Fourier expansion	Notes	6.2-6.4
28	Dec. 6	Operators on Hilbert spaces, partial differential equations	Notes	7.1-7.2, 8.1-8.3