**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.