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
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
determine whether or not specified functions of
several real variables satisfy differentiability
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.
Principles of Real Analysis, W. Rudin. McGraw-Hill, 1976;
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.