# Analysis III

In this third course in the analysis sequence, we study the calculus of functions of two or three variables and interpret the fundamental theorems of calculus in higher dimensions, and which have rich applications in the physical sciences.

This last course in analysis deals with the calculus of functions of two and three real variables. The learners in earlier courses of analysis are equipped with the skills of visualising such functions and this course builds on it to understand the notions of differentiation and integration of these functions as limiting processes, by motivating the key ideas through their geometric interpretations.

The first part of the course discusses the notion of differentiability of functions from R^n (n = 2,3) to R and the gradient first, and then the notion of differentiability of functions from R to R^n (n = 2,3). It also discusses important properties such as the Inverse function theorem, Implicit function theorem, and the generalised Taylor’s theorem, and their applications involving modelling.

The second part of the course studies double and triple integrals, their applications, and some numerical methods for computing them.

The last part of the course discusses line and surface integrals, their computations, and important properties such as Gauss’ theorem, Green’s theorem and their consequences.