# Module MA2321: Analysis in several real variables

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2018-19
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof. David Wilkins
Learning Outcomes
On successful completion of this module, students will be able to:
• justify with logical argument basic results concerning the topology of Euclidean spaces, convergence of sequences in Euclidean spaces, limits of vector-valued functions defined over subsets of Euclidean spaces and the continuity of such functions;
• specify accurately the concepts of differentiability and (total) derivative for functions of several real variable;
• justify with logical argument basic properties of differentiable functions of several real variables including the Product Rule, the Chain Rule, and the result that a function of several real variables is differentiable if its first order partial derivatives are continuous;
• determine whether or not specified functions of several real variables satisfy differentiability requirements.
Module Content
• Review of real analysis in one real variable: the real number system; the least upper bound axiom; convergence of monotonic sequences; the Bolzano-Weierstrass Theorem in one variable; the Extreme Value Theorem; differentiation; Rolle's Theorem; the Mean Value Theorem; Taylor's Theorem; the Riemann integral.
• Analysis in several real variables: convergence of sequences of points in Euclidean spaces; continuity of vector-valued functions of several real variables; the Bolzano-Weierstrass Theorem for sequences of points in Euclidean spaces; the Extreme Value Theorem for functions of several real variables.
• Differentiablity for functions of several variables: partial and total derivatives; the Chain Rule for functions of several real variables; properties of second order partial derivatives.
Module Prerequisite
MA1111 Linear Algebra 1, MA1125 Single-Variable Calculus and Introductoy Analysis, MA1132 Advanced Calculus