# Mathematical Physics

B.Sc. pt II (hons) course

Syllabus | In classes | Links | References

**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 August 3,

new course material from August 19, 2016

Reference Books: Mathematical Methods for Physics and Engineering by Riley-Hobson-Bence (Cambridge University Press)

## In class

**Revision Lecture 0.1, 0.2, 0.3: Curve tracing and their use in Physics**: 3, 4 and 5 August 2016**Revision Lecture 0.4, 0.5, 0.6: Differentiation, integration, their gemotrical meaning and uses**: 10, 11 and 12 August 2016**No student was present**: 17 August 2016**Holiday: Rakshabandhan**: 18 August 2016**Lecture 1, 2: Introduction to Curvilinear co-ordinate system**: 19 August 2016**Lecture 3: Scale factors**: 24 August 2016**Holiday: Janmashtami**: 25 August 2016**Lecture 4, 5: Introduction to cylindrical and spherical co-ordinates**: 26 August 2016**Student election:**: 31 August 2016**Holiday: Student president power**: 1 September 2016**All students were absent.**: 2 September 2016**Lecture 6: Gradient and its meaning**: 7 September 2016**Lecture 7: Divergence and Curl**: 8 September 2016**Lecture 8, 9: Numerical problems in curvilinear**: 9 September 2016**Lecture 9:**: 14 September 2016**Holiday: Anant Chaturdashi (Hindu and Jain festival)**: 15 September 2016**Lecture 10, 11: Co-ordinate transformation and Jacobian**: 16 September 2016**Lecture 12: Covariant, Contravariant and Mixed Tensors**: 21 September 2016**Lecture 13: Algebra of tensors**: 22 September 2016**Lecture 14, 15: Metric tensor and its conjugate tensor**: 23 September 2016**Lecture 16: Transformation of tensors**: 28 September 2016**Lecture 17: Dirac Delta Function**: 29 September 2016**Lecture 18, 19: Properties of Dirac Delta Function**: 30 September 2016**Lecture 20: Functions and their properties**: 5 October 2016**Lecture 21: Introduction to Fourier Series**: 6 October 2016**Lecture 22, 23: Computation of Fourier coefficients**: 7 October 2016**Holiday: Muharram (Muslim festival)**: 12 October 2016**Lecture 24: Even and odd functions**: 13 October 2016**Lecture 25, 26: Half range expansion**: 14 October 2016**Lecture 27: Problem solving**: 19 October 2016**All students were absent.**: 20 October 2016**Special Holiday: University holiday**: 21 October 2016**Special Holiday: University holiday**: 26 October 2016**Special Holiday: University holiday**: 27 October 2016**Holiday: Dhanteras (Hindu festival)**: 28 October 2016**Lecture 28: Problem solving**: 2 November 2016**Lecture 29: Geometrical interpretation**: 3 November 2016**Lecture 30, 31: Sums and scale changes, forced oscillations**: 4 November 2016**Lecture 32: Expansion Techniques: integration and differentiation**: 9 November 2016**Lecture 33: Fourier series in exponential form**: 10 November 2016**Lecture 34, 35: Introduction to Fourier transform**: 11 November 2016**Lecture 36: Some examples of Fourier transform**: 16 November 2016**Lecture 37: Dirac delta function in integration of exponential**: 17 November 2016**Lecture 38, 39: Problem solving on FT**: 18 November 2016**Lecture 40: RENF and rank of a matrix**: 23 November 2016**Lecture 41: Solution of linear equations using matrix**: 24 November 2016**Lecture 42, 43: Normas and inner products**: 25 November 2016**Lecture 44: Orthogonal sets and matrices**: 31 November 2016**Lecture 45: Gram Schmidt process and Q-R factorization of matrices**: 1 December 2016**Lecture 46, 47: Problem solving**: 2 December 2016**Lecture 48: Projection matrices**: 7 December 2016**Lecture 49: Least square fit of data**: 8 December 2016**Lecture 50, 51: Eigen values and Eigen vectors**: 9 December 2016**Lecture 52: Diagonalization of matrices**: 14 December 2016**Lecture 53: Real symmetric and simple matrices**: 15 December 2016**Lecture 54, 55: Linear differential equations for the homogeneous and non-homogeneous cases**: 16 December 2016**Lecture 56: Solution of differential equations**: 21 December 2016**Science exhibition: teaching off**: 22 December 2016**Science exhibition: teaching off**: 23 December 2016

There were classes in January 2017 too, couldn't get updated on the website.

## Syllabus

### 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 for Physics and Engineering by Riley-Hobson-Bence (Cambridge University Press)
- 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.

## Links:

**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**