MAU23203 Analysis in several real variables

Module Code	MAU23203
Module Title	Analysis in several real variables
Semester taught	Semester 1
ECTS Credits	5
Module Lecturer	Prof. David Wilkins
Module Prerequisites	MAU11101 Linear algebra I, MAU11202 Advanced calculus and MAU11204 Analysis on the real line

11 weeks of teaching with 3 lectures per week.

On successful completion of this module, students will be able to

Justify with logical arguments basic results concerning the convergence of sequences in Euclidean spaces and the continuity of vector-valued functions defined on subsets of Euclidean spaces.
Expound significant aspects of the theory of differentiability, as it applies to real-valued functions of several real variables.
Discuss in substantial detail situations where standard properties of partial derivatives of first and second order satisfied by smooth functions fail to extend to situations where the partial derivatives of first or second order fail to satisfy appropriate continuity conditions.
Analyze mathematical configurations specified using the language of convergence and continuity, as it is applied to sequences in Euclidean spaces, and to functions between subsets of Euclidean spaces, so as to construct reasoned logical arguments intended to justify stated properties of such configurations.

Review of real analysis in one real variable: the real number system, the least upper bound axiom, convergence of monotonic sequences, Bolzano-Weierstrass theorem in one variable, extreme value theorem, differentiation, Rolle's theorem, mean value theorem, Taylor's theorem, aspects of the theory of integration.
Analysis in several real variables: convergence of sequences of points in Euclidean spaces, continuity of vector-valued functions of several real variables, Bolzano-Weierstrass theorem for sequences of points in Euclidean spaces, extreme value theorem for functions of several real variables.
Differentiability for functions of several variables: partial and total derivatives, chain rule for functions of several real variables, properties of second order partial derivatives, inverse function theorem, implicit function theorem.