Introductory description

Mathematical Analysis is the heart of modern Mathematics. This module is the final in a series of modules where the subject of Analysis is rigorously developed in many dimensional setting.

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

• Different notions of continuity of functions of several variables
• Quantitative Linear Algebra in terms of norms
• Different notions of differentiability of functions of several variables
• Chain rule, (generalised) mean value inequality and other properties of differentiable functions
• Inverse Function Theorem and Implicit Function Theorem, with applications to regular curves and hypersurfaces
• Vector Fields and the theorems of Green, Gauss and Stokes, with some applications to PDEs.
• Maxima, minima and saddles and constrained critical points.

Learning outcomes

By the end of the module, students should be able to:

• learn the basic concepts, theorems and calculations of multivariable analysis
• understand the Implicit and Inverse Function Theorems and their applications
• acquire a working knowledge of vector fields and the Integral Theorems of Vector Calculus
• learn how to analyse and classify critical points using Taylor expansions

Indicative reading list

J. E. Marsden and A. Tromba. Vector Calculus. Macmillan Higher Education, sixth edition, 2011.
J. J. Duistermaat, J. A. C. Kolk. Multidimensional Real Analysis I : Differentiation, CUP, 2004 [available online via Warwick's library]
R. Coleman. Calculus on normed vector spaces, Springer 2012. [available online via Warwick's library]
W. Rudin. Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, third edition, 1976.
T. M. Apostol. Mathematical Analysis. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., second edition, 1974.
T. W. Körner. A Companion to Analysis: A Second First and First Second Course in Analysis, volume 62 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2004.

Subject specific skills

Multivariable Analysis gives students tools to do rigorous Analysis in higher dimensional spaces. Students will learn definitions, theorems and calculations with vector-valued functions of many variables, for instance, Inverse and Implicit Function Theorems, vector fields, maxima, minima and saddles.

Transferable skills

Students will acquire key reasoning and problem solving skills, empower them to address new problems with confidence.