2007 Handbooks

This unit deals with two of the most fundamental concepts in analysis -- complex analysis and vector analysis.

Complex analysis is the study of complex-valued functions of complex variables. Two approaches to the study of complex-valued functions of one complex variable are discussed. The first of these, usually attributed to Riemann, is based on differentiation and involves pairs of partial differential equations called the Cauchy-Riemann equations. The second approach, usually attributed to Cauchy, is based on integration and depends on a fundamental theorem known nowadays as Cauchy's integral theorem.

The concept of vector analysis provides the tools for modelling physical phenomena such as fluid flow, electromagnetic and other field-based theories. We consider vector fields and integrals over paths and surfaces, and develop an understanding of the famous integration theorems of Green, Stokes and Gauss. These theorems transform physical laws expressed in differential form to integral form.