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Module MA2321: Analysis in several real variables

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2016-17
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
 
Recommended Reading
  • Principles of Real Analysis, W. Rudin. McGraw-Hill, 1976;
 
Assessment Detail
This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session, with the examination counting for the remaining 90%. The supplemental examination paper, if required, will determine 100% of the supplemental module mark.