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Complex Variables : Regions and paths in the Z plane. Path/Line integral of a function.

Inequality conditions for a path integral to be independent of the path joining two points. Contour Integral, Cauchys theorem for analytical functions with continuous derivatives. Cauchy Goursat theorem( statement only ) and its use for multiply connected regions. Cauchys integral formula and deductions. Moreras theorem and maximum modulus theorem. Taylors and Laurents developments, Singularities, poles, residue at isolated singularity and its evaluation. Residue theorem - Application to evaluate real integrals.

Matrices : Brief revision of vectors over real field, inner product, normal, linear independence, orthogonality. Characteristic values and vectors, and their properties for Hermitian and real Symmetric matrices : Characteristic polynomial. Cayley Hamilton theorem, functions of a square matrix, minimal polynomial, diagonable matrix.

Quadratic forms, orthogonal, congruent and Lagranges reduction of quadratic forms, rank, index, signature of a quadratic form, value class of a quadratic form. Statement of bilinear form.

Vector Calculus : Scalar and Vector point functions, directional derivative, level surfaces, gradient, surface and volume integrals, definition of curl, divergence. Use of operator. Conservative, irrotational, solenoidal fields. Greens theorem for plane regions and properties of line integral in a plane, Statements of Stokes theorem, Gauss Divergence theorem, related identities, deductions, statement of Laplaces differential equation in cartesian, spherical, polar and cylindrical co-ordinates.

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