# Mathematical Physics

B.Sc. pt II (hons) course

Syllabus | In Next class | In prevoius classes | Links

**Dates and timings: **

Wednesday and Thursday, 12:00 PM – 01:00 PM (IST)

Friday, 10:00 AM – 12:00 PM (IST)

**Starting date: **

Revision of pre-requisites from July 29,

new course material from August 7, 2015

**No student was present**: 29 and 30 July 2015**Revision Lecture 0.1, 0.2, 0.3: Relation of physics and mathematics**: 31 July, 5 and 6 August 2015**Lecture 1: Introduction to Matrices**: 7 August 2015**Lecture 2: Algebra of Matrices**: 12 August 2015**Lecture 3: Inverse of Matrices, adjoint**: 13 August 2015**Lecture 4, 5: Solution of linear equations using matrices**: 14 August 2015**Lecture 6: Orthogonal sets of matrices**: 19 August 2015**Lecture 7: Gram Schmidt process**: 20 August 2015**Lecture 8, 9: Q-R factorization theorem**: 21 August 2015**Students election:**: 26 August 2015**Holiday:**: 27 August 2015**Lecture 10, 11: Projection matrices and some examples**: 28 August 2015**Lecture 12: Theorems related to projection matrices**: 2 September 2015**Lecture 13: Least square fit of data**: 3 September 2015**Lecture 14, 15: Matrices as operator**: 4 September 2015**Lecture 16: Introduction to linear independence**: 9 September 2015**Lecture 17: Eigen values**: 10 September 2015**Lecture 18, 19: Eigen values and eigen vectors**: 11 September 2015**Lecture 20: Diagonalization of matrices**: 16 September 2015**Holiday:**: 17 September 2015**Holiday:**: 18 September 2015**Holiday:**: 23 September 2015**Lecture 21: Orthogonal Curvilinear co-ordinate system**: 24 September 2015**Holiday:**: 25 September 2015**Lecture 22: Scale factors, expression for gradient**: 30 September 2015**Lecture 23: Divergence and curl**: 1 October 2015**Lecture 24, 25: Cartesian, cylindrical and spherical co-ordinate systems**: 2 October 2015**Lecture 26: Co-ordinate transformation and Jacobian**: 7 October 2015**Lecture 27: Covariant, Contravariant and Mixed Tensors and their mathematics**: 8 October 2015**Lecture 28, 29: Metric tensor and uses, Dirac Delta Function and its properties**: 9 October 2015**Lecture 30: Introduction to Fourier series**: 14 October 2015- References: Hassani, 2nd edition, Page 299-303
**Lecture 31: Fourier theorem and computation of Fourier coefficients**: 15 October 2015- References: Hassani, 2nd edition, Page 299-303
**Lecture 32, 33: Sums and scale changes in Fourier series**: 16 October 2015- References: Potter-Goldberg
**Holiday**: 21 October 2015**Holiday**: 22 October 2015**No student was present**: 23 October 2015**Lecture 34: Fourier series: Even and odd functions, range expansion, forced oscillations**: 28 October 2015**Lecture 35: Introduction to Fourier transform and its simple applications**: 29 October 2015**Lecture 36, 37: Expansion Techniques: integration and differentiation and problems of Fourier series**: 30 October 2015**Holiday**: 4 November 2015**Holiday**: 5 November 2015**Holiday**: 6 November 2015**Holiday**: 11 November 2015**Holiday**: 12 November 2015**Holiday**: 13 November 2015**Lecture 38: Solution of differential equations - Series method, Properties of power series**:18 November 2015**Lecture 39: Legendre's Equation**: 19 November 2015**Lecture 40, 41: Legendre Polynomials and Functions**: 20 November 2015**Holiday**: 25 November 2015**Lecture 42: Hermite Polynomials**: 26 November 2015**Lecture 43, 44: The method of Frobenius: Solution about regular singular points, The Gamma function**:27 November 2015**Lecture 45: The Bessel-Clifford equation, Solution of Bessel equation**: 02 December 2015**Lecture 46: Solution of Bessel equation**: 03 December 2015**Lecture 47, 48: Identities and properties of Bessel, Hermite and Legendre and associated Legendre's functions**:04 December 2015**Lecture 49: Simple applications of above functions**:09 December 2015**Lecture 50: Problem solutions and Rodrigues formulas**:10 December 2015**Lecture 51, 52: Solution of partial differential by separation of variable technique.**:11 December 2015**Lecture 53: Laplace, Poisson and Helmholtz equation. in two and three dimensional and its applications**:16 December 2015**Lecture 54: Solution of Laplace equation in 1, 2 and 3D.**: 17 December 2015**No student was present (some function in college)**: 18 December 2015**Lecture 55: Some applications of Laplace equation**: 23 December 2015**Lecture 56: Laplace, Helmholtz and Poisson equations in different coordinate systems.**: 6 January 2016**Lecture 57: Solution of Laplace equation in different coordinate systems.**: 7 January 2016**Lecture 58, 59: Problem solution for line charge between two earthed parallel plates**: 8 January 2016**Lecture 60: Applications of Laplace equation**: 13 January 2016**Lecture 61: Solution of linear differential equations in matrices**: 14 January 2016**Holiday**: 15 January 2016

### Unit I

Orthogonal Curvilinear co-ordinate system. Scale factors, expression for gradient, divergence and curl and their applications to Cartesian, circular, cylindrical and spherical polar co-ordinate systems.

Co-ordinate transformation and Jacobian. Transformation of Covariant, Contravariant and Mixed Tensors. Addition, Multiplication and Contraction of Tensors. Metric tensor and its use in transformation of tensors. Dirac Delta Function and its properties.

### Unit II

Fourier series: Fourier theorem and computation of Fourier coefficients. Even and odd functions, half range expansion, sums and scale changes, forced oscillations, Expansion Techniques: integration and differentiation. Introduction to Fourier transform and its simple applications.

### Unit III

Solution of differential equations - Series method: Properties of power series, solution of ordinary differential equation: Legendre's Equation, Legendre Polynomials and Functions, Hermite Polynomials.

The method of Frobenius: Solution about regular singular points, The Gamma function, the Bessel-Clifford equation.

Roots differing by an integer: Series method, Solution of Bessel equation for:

- Roots not differing by an integer
- Equal roots
- Roots differing by an integer

Basic identities involving Bessel Functions. Basic properties like orthogonality recurrence relation and generating functions of Bessel, Hermite, Legendre, and associated Legendre's function (simple applications)

### Unit IV

Solution of partial differential by separation of variable technique and its application to following Boundary Value Problems:

- Laplace equation in three dimensional Cartesian co-ordinate system - line charge between two earthed parallel plates.
- Laplace equation in two dimensional Cartesian co-ordinate system - Heat conduction in a thin rectangular plate.
- Wave equation in two dimensional Cartesian co-ordinate system - Heat conduction in a thin rectangular plate.
- Diffusion equation in cylindrical co-ordinate system.

### Unit V

Matrices: Inverse of a matrix, adjoint, Hermition adjoint, Solution of linear equations using matrix.

Normas and inner products, orthogonal sets and matrices, the Gram Schmidt process and the Q-R factorization theorem. Projection matrices. Least square fit of data. Eigen values and Eigen vectors, diagonalization of matrices. Examples involving up to 3×3 matrices and for the case of real symmetric and simple matrices. Solution of linear differential equations for the homogeneous and non-homogeneous cases.

**Reference books**

- Mathematical Methods by Potter and Glodberg (Prentice Hall of India Pvt. Ltd.) (I followed.)
- Applied Mathematics for Engineers and Physicists by Pipes and Harvill (McGraw Hill Book Co.)
- Mathematical Methods for Physicists (6ed) by George B. Arfken and Hans J. Weber.

**On wikipedia**

- Curvilinear coordinates
- Tensor
- Dirac delta function
- Fourier series
- Fourier transform
- Differential equation
- Legendre polynomials
- Hermite polynomials
- Gamma function
- Bessel–Clifford function
- Partial differential equation
- Laplace's equation
- Diffusion equation
- Matrix / Matrices
- Eigenvalues and eigenvectors

**Other sites**

**ocw.mit.edu**