Double integration (Cartesian co-ordinates) – Change of order of integration – Applications of double integral (Area) – Triple Integration (Cartesian co-ordinates) – Applications: Volume as triple integrals and solids of revolution.

Derivatives: Gradient of a scalar field. Directional derivative – Divergence of a vector field – Curl of a vector field – Solenoidal and Irrotational of a vector – Theorems in divergence of Gauss, Stoke’s and Greens (statements only) – Verification of theorem.

Derivatives of f(z) – Analytic function – Cauchy-Riemann Equations – Harmonic function – Harmonic conjugate – Construction of Analytic function – Conformal Mapping – Conformality of w= z + c, cz , 1/z – Mobius transformations – Application to flow problems.

Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s series – Zeros of an analytic function – singularities – Laurent’s series – Residues – Cauchy Residue theorem.

Conditions – Transforms of elementary functions – Properties - Transform of derivatives and integrals – Unit step function (Heaviside function) –Dirac’s Delta function – Laplace transform of periodic functions – Inverse Laplace transforms - Convolution theorem – Partial fraction method – Applications to solution of linear ordinary differential equations of second order with constant coefficients – Simultaneous linear equations with constant coefficients.