Vector Calculus and Complex Variable

Learning Outcomes

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Discover how calculus of real functions can be extended to complex functions, thus developing a completely new perspective on many seemingly unrelated aspects of mathematics covered in other modules, e.g complex Fourier series for solving partial differential equations or stability of approximation methods in numerical analysis.
• Develop an understanding of how calculus, in both its differential and integral aspects, can be seamlessly extended from scalar functions of one variable to vector functions of many variables.

Learning Outcomes

Having successfully completed this module you will be able to:

• Use Gauss’s and Stokes’ theorems to simplify calculations of integrals and prove simple results;
• Evaluate line, surface and volume integrals in simple coordinate systems;
• Calculate grad, divergence and curl in Cartesian and other simple coordinate systems, and establish identities connecting these quantities;
• Parametrize a path and evaluate some complex integrals directly;
• Evaluate real and complex integrals using the Cauchy integral formula and the residue theorem.

PART A: VECTOR CALCULUS

Review of basic vector algebra

Differentiation of vector valued functions of one variable

Vector fields

Parametrization of curves and line integrals

Parametrization of surfaces and flux integrals

Gradient, divergence and curl; basic theory and simple examples in Cartesian coordinates

Identities in vector calculus

Integral theorems; Gauss, Stokes and Green’s theorems

Scalar potential, vector potential

General orthogonal curvilinear coordinates

PART B: COMPLEX VARIABLE THEORY

Review of analytic functions and Cauchy-Riemann equations

Paths in the complex plane, parametrization, contours

Contour integrals

Cauchy’s theorem

Cauchy integral formula

Liouville’s theorem and fundamental theorem of algebra

Laurent and Taylor series

Singularities, residues and the residue theorem

Evaluation of real integrals

Lectures, problem classes, private study

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.