| Indicative Syllabus • Vector Calculus • Review of vector basics, applications of vectors in physical examples. • Develop scalar and vector triple product formulae. • Functions of several variables. Geometric interpretation. Level Curves. • Partial derivatives. Geometric interpretation of the partial derivative. • Chain rules for function of several variables. • Scalar and vector functions. Physical examples. • Introduce the "Del" vector differential operator. • The gradient of a scalar field. r-dependent vector and scalar fields. Physical examples. • Directional Derivative. Examples involving steepest ascent/descent. Directions of zero change. • Divergence of a vector field. • The Laplacian operator. Poisson's and Laplace's equation. • The Curl of a vector field. Conservative vector fields. • Discussion of Maxwell's equations. • Derivative of vector functions. Unit tangent, normal, and binormal vectors. • Curvature, radius of curvature and torsion. • The Frenet-Serret formulae. • Velocity and acceleration vectors vectors. • Line integrals. • Path independence. Scalar potential of a conservative vector function. • Complex Analysis • Basic concepts of complex numbers. Functions of a complex variable. • The Cauchy-Riemann equations. Analytic functions. • Harmonic functions. Connection between harmonic functions and analytic functions. • Solving equations in complex variables. • Integration of a complex function and line integrals in the complex plane. • Cauchy's Integral theorem. • The Cauchy Integral Formula and contour integrals. • The Cauchy Integral Formula for derivatives. • Laurent Series and the Residue theorem. |