MA26310 Multivariable Analysis
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
extend the analysis of one variable from the first year to the multivariable context.
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. McGrawHill Book Co., New YorkAucklandDüsseldorf, third edition, 1976.
T. M. Apostol. Mathematical Analysis. AddisonWesley Publishing Co., Reading, Mass.LondonDon 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 vectorvalued 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.
Study time
Type  Required 

Lectures  20 sessions of 1 hour (20%) 
Online learning (independent)  9 sessions of 1 hour (9%) 
Private study  13 hours (13%) 
Assessment  58 hours (58%) 
Total  100 hours 
Private study description
Working on assignments, going over lecture notes, text books, exam revision.
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Assignments  15%  20 hours 
Examination  85%  38 hours 
Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100% 
Feedback on assessment
Marked homework (both assessed and formative) is returned and discussed in smaller classes. Exam feedback is given.
Courses
This module is Core for:
 Year 2 of UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)

UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)
 Year 2 of G103 Mathematics (MMath)
This module is Core optional for:

UMAAG100 Undergraduate Mathematics (BSc)
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 2 of G100 Mathematics
 Year 2 of UMAAG103 Undergraduate Mathematics (MMath)
 Year 2 of UMAAG1NC Undergraduate Mathematics and Business Studies
 Year 2 of UMAAG1N2 Undergraduate Mathematics and Business Studies (with Intercalated Year)
 Year 2 of UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Option list A for:
 Year 2 of UMAAGL11 Undergraduate Mathematics and Economics
 Year 2 of UECAGL12 Undergraduate Mathematics and Economics (with Intercalated Year)

UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 2 of GF13 Mathematics and Physics
 Year 2 of GF13 Mathematics and Physics

UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
 Year 2 of FG31 Mathematics and Physics (MMathPhys)
 Year 2 of FG31 Mathematics and Physics (MMathPhys)

USTAGG14 Undergraduate Mathematics and Statistics (BSc)
 Year 2 of GG14 Mathematics and Statistics
 Year 2 of GG14 Mathematics and Statistics
This module is Option list B for:
 Year 2 of UCSAG4G1 Undergraduate Discrete Mathematics
 Year 2 of UCSAG4G3 Undergraduate Discrete Mathematics

USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
 Year 2 of Y602 Mathematics,Operational Research,Stats,Economics
 Year 2 of Y602 Mathematics,Operational Research,Stats,Economics