# Analysis II

Continuing our study of real numbers and their properties, we study functions on reals and their properties through some theory of differentiation and integration. We will also study how these two concepts are related.

This is the second course in the sequence of analysis courses. The first part of the course introduces the notion of differentiation as a limiting process and discusses several properties of functions from R to R, through this process. The major properties include the Mean Value Theorem, Taylor’s theorem and the Inverse function Theorem, and their consequences.

The second part of the course deals with integration of real functions, series of real numbers, and their inter-relation. The major properties here include the Fundamental theorems of calculus and their consequences. Numerical methods of integration are also discussed.

The last part of the course is a precursor to the study of functions of two and three variables, which are studied in Analysis 3. Here, the structure of the two-dimensional (R^2) and the three-dimensional (R^3) spaces, and sequences in R^2, R^3 are discussed thoroughly. The course ends with the notions of level sets, curves and surfaces and their visualisations.