CBCGS [2017 - current]

CBGS [2013 - 2016]

Old [2000 - 2012]

## Topics with syllabus and resources

1 Module 1

1.01 Calculus of Variation

- Euler Langrange equation.
- Solution of Euler‘s Langrange equation (only results for different cases for function) independent of a variable, independent of another variable, independent of differentiation of a variable and independent of both variables.
- Isoperimetric problems, several dependent variables.

1.02 Functions Involving Higher Order Derivatives

- Rayleigh-Ritz method

2 Module 2

2.01 Linear Algebra: Vector Spaces

2.02 Vectors in N-dimensional Vector Space

- Properties
- Dot product
- Cross product
- Norm and distance properties in n-dimensional vector space.
- Metric spaces, vector spaces over real field
- Properties of vector spaces over real field, subspaces.
- Norms and normed vector spaces.
- Inner products and inner product spaces
- The Cauchy-Schwarz inequality, orthogonal Subspaces
- Gram-Schmidt Process.

3 Module 3

3.01 Linear Algebra: Matrix Theory

- Characteristic equation
- Eigenvalues and Eigenvectors
- Properties of Eigenvalues and Eigenvectors
- Cayley-Hamilton theorem, examples based on verification of CayleyHamilton theorem.
- Similarity of matrices
- Diagonalisation of matrix Functions of square matrix
- Derogatory and non-derogatory matrices Quadratic forms over real field
- Reduction of quadratic form to a diagonal canonical form
- rank, index, signature of quadratic form
- Sylvester‘s law of inertia
- value-class of a quadratic form of definite, semidefinite and indefinite Singular Value Decomposition.

4 Module 4

4.01 Complex Integration

- Line Integral
- Cauchy‘s Integral theorem for simply connected regions
- Cauchy‘s Integral formula
- Taylor‘s and Laurent‘s series.

4.02 Complex Variables

- Zeros
- Singularities
- Poles of f(z)
- Residues
- Cauchy‘s Residue theorem
- Applications of Residue theorem to evaluate real Integrals of different types.