Vector fields: Orthogonal curvilinear co-ordinate systems – Expressions for gradient,divergence, curl and Laplacian- Linear vector spaces: Linear independence, basis, dimension,inner product – Schwartz inequality – Orthonormal basis – Gram – Schmidt orthogonalization process –Linear operators – Representation of vectors and operators in a basis – Matrix theory – Cayley - Hamilton theorem – Inverse of a matrix – Diagonalisation of matrices – Operational methods: Laplace transforms – Solution of linear differential equations with constant coefficients – Fourier integral –Fourier transforms – Convolution theorems – Applications – Complex variables: Analytic function – Cauchy – Riemann conditions – Singular points - Multivalued function and branch points – Cauchy’s integral theorem and formula – Taylor’s and Laurentz’s expansions - Residue theorem and its applications.
