On successful completion of this module, students will be able to:
justify with logical argument basic results concerning convergent sequences in Euclidean spaces and of vector-valued functions defined on subsets of Euclidean Spaces.
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;
exploit the Inverse Function Theorem and related
results to identify and justify the basic properties
of smooth submanifolds of Euclidean spaces.
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; the Inverse Function Theorem; the Implicit Function Theorem; curvilinear coordinate systems.
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.