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.