
MA2321  Analysis in Several Real Variables
Dr. David R. Wilkins
Course Notes  Michaelmas Term 2016

Course Notes for Michaelmas Term 2016

Part I (Sections 1, 2 and 3) includes the following sections.
 Section 1: The Real Number System
 This section begins by reviewing the characterization of the
real number system as an ordered field satisfying the
Least Upper Bound Property. Bounded monotonic
sequences are shown to be convergent, and the statement
and proof of the BolzanoWeierstrass Theorem
in one dimension are given. The formal definition of
continuity for functions of a real variable is introduced,
and the Intermediate Value Theorem and the
Extreme Value Theorem are stated and proved
(in the onedimensional case). Continuous realvalued
functions on closed bounded intervals are shown to be
uniformly continuous.
 Section 2: The Mean Value Theorem
 The definition of differentiability is given, and the
Product Rule, Quotient Rule and Chain Rule and the
derivatives of standard trigonometrical, logarithm and
exponential functions are briefly stated without proof.
A proof of Rolle's Theorem is given, making use
of the Extreme Value Theorem for continuous functions
of a single real variable. The Mean Value Theorem
is proved. A special case of Taylor's Theorem
for twicedifferentiable functions is proved, and is then
applied to prove that those stationary points of a
differentiable function of one real variable where the
second derivative is positive are local minima.
 Section 3: The Riemann Integral on One Real Variable
 The Riemann Integral (or RiemannDarboux Integral
is defined, using Darboux upper and lower sums, and its
basic properties are derived. Sums and differences of
Riemannintegrable functions are shown to be Riemannintegrable.
It is also shown that the absolute values of a Riemannintegrable
function yield a Riemannintegrable function. Products of
Riemannintegrable functions are shown to be Riemannintegrable.
Monotonic functions on closed bounded intervals are shown to
be Riemannintegrable. Also the result that all
continuous realvalued functions are uniformly continuous
on closed bounded intervals is applied to prove that
continuous functions are Riemannintegrable. The
Fundamental Theorem of Calculus is proved,
and is then used to justify the rules for
Integration by Substitution and
Integration by Parts. An example is given to
show that the integral of a pointwise limit of a
sequence of polynomial functions on a closed bounded
interval is not always the limit of the integrals of
those polynomial functions. The concept of uniform
convergence is introduced, and it is shown that, for
a uniformly convergent sequence of continuous realvalued
functions, the limit function is continuous. It is
also shown that limits and integrals may be interchanged
in cases involving the uniform convergence of a sequence
of Riemannintegrable functions to a Riemannintegrable
function.

Part II (Sections 4, 5, 6 and 7) includes the following sections.
 Section 4: Continuous Functions of Several Real Variables
 Section 5: Compact Subsets of Euclidean Spaces
 Section 6: The Multidimensional RiemannDarboux Integral
 Section 7: Norms on FiniteDimensional Euclidean Spaces

Part III (Sections 8 and 9) includes the following sections.
 Section 8: Differentiation of Functions of Several Real Variables
 Section 9: The Inverse and Implicit Function Theorems

Part IV (Sections 10 and 11) includes the following sections.
 Section 10: Second Order Partial Derivatives and the Hessian Matrix
 Section 11: Repeated Differentiation and Smoothness
Back to D.R. Wilkins: MA2321
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.