EM 217: Advanced Calculus

Review of partial derivatives, geometric interpretation, total differentials, and chain rules, sketching level curves and level surfaces of functions of two and three variables, sketching surfaces and solids, limit, and continuity of functions of two and three variables, tangent planes and linear approximation, directional derivative and gradient vector.

Mean value theorem, Taylor’s theorem, extreme value theorem and second derivative test, Lagrange multipliers, scalar line integrals.

Definitions of double and triple integrals, double and triple integrals over rectangular domains, double and triple integrals over any general domains, cylindrical and Spherical polar coordinates, Jacobian and its properties, applications of double and triple integrals. (change of coordinates)

Scalar fields and vector fields, gradient, Divergence and Curl and their geometrical and physical interpretations.

Line integrals of vector valued functions and path independency of line integrals, simply connected domains and conservative vector fields, Cauchy-Riemann equations and Line integrals of complex-valued functions, complex line integrals over simply connected domains and Cauchy’s Theorem, applications of harmonic functions.

Greens Theorem on the plane, surface integrals of scalar fields and vector fields, Stokes’ theorem and divergence theorem, area and volume elements in terms of orthogonal curvilinear coordinates, surface integrals with orthogonal curvilinear coordinates, applications of integral theorems in terms of orthogonal curvilinear coordinates.