
MA2321  Analysis in Several Real Variables
Dr. David R. Wilkins
Lecture Slides  Michaelmas Term 2017

Lecture Material
 Lecture 1 (September 25, 2017)
 This lecture covered the definition of an
ordered field, the Least Upper Bound Principle,
and the characterization of the
real number system as a Dedekindcomplete ordered
field.
 Lecture 2 (September 28, 2017)
 This lecture covered basic results concerning
convergence of infinite sequences of real numbers,
including the onedimensional case of the
BolzanoWeierstrass Theorem.
 Lecture 3 (September 28, 2017)
 This lecture covered basic results concerning
the scalar product and Euclidean norm of vectors
in ℝ^{n}, including
Schwarz's Inequality and the Triangle Inequality.
The definition of convergence was presented for
sequences of points in Euclidean spaces
ℝ^{n} of arbitrary (finite)
dimension n. The lecture also presented
the definitions of limit points and
isolated points of subsets of
ℝ^{n}.
 Lecture 4 (October 2, 2017)
 This lecture covered versions of the
BolzanoWeierstrass Theorem and Cauchy's Criterion
for Convergence applicable to bounded sequences
of points in Euclidean spaces ℝ^{n}
of arbitrary (finite) dimension n.
 Lecture 5 (October 5, 2017)
 This lecture introduced the definition of open sets
in Euclidean spaces, and in subsets of Euclidean spaces,
and developed basic properties of the collection of
open sets in subsets of Euclidean spaces.
 Lecture 6 (October 5, 2017)
 This lecture presented a criterion for convergence
for a sequence of points in a Euclidean spaces, expressed
in terms of the open sets of that space, and not explicitly
referencing the distance function on that space. The
lecture continued with a discussion of closed sets in
subsets of Euclidean spaces.
 Lecture 7 (October 9, 2017)
 This lecture introduced the basic definitions of
limits and continuity for functions
of several real variables, and developed some basic
consequences of those definitions.
 Lecture 8 (October 12, 2017)
 This lecture developed further the theory of
limits of functions of several real variables.
 Lecture 9 (October 12, 2017)
 This lecture developed further the theory of
continuous functions of several real variables.
The lecture also presented applications of the
multidimensional BolzanoWeierstrass Theorem to prove
various results concerning continuous functions on closed
bounded subsets of Euclidean spaces, including a
multidimensional version of the Extreme Value Theorem,
the result that continuous functions defined over
such closed bounded sets are uniformly continuous,
and a proof of the equivalence of norms on a
finitedimensional vector space.
 Lecture 10 (October 19, 2017)
 This lecture presented the definition of the Riemann
integral for functions of a single real variable,
adopting the approach using Darboux upper and
lower sums.
 Lecture 11 (October 19, 2017)
 This lecture developed the theory of the Riemann
integral for functions of a single real variable.
 Lecture 12 (October 23, 2017) (not final)
 This lecture discussed partitions of closed
ndimensional cells in
ℝ^{n}
 Lecture 13 (October 26, 2017) (not final)
 This lecture developed the definition of the Riemann
integral for functions of several real variables.
 Lecture 14 (November 2, 2017) (not final)
 This lecture developed the definition of the Riemann
integral for functions of several real variables.
 Lecture 15 (November 2, 2017) (not final)
 This lecture developed the theory of the Riemann
integral for functions of several real variables.
 Lecture 16 (November 13, 2017) (not final)
 This lecture reviewed the basic theory of
differentiation, for functions of one real variable.
 Lecture 17 (November 16, 2017) (not final)
 This lecture included a proof of Taylor's Theorem (with remainder),
and also applied the Mean Value Theorem in conjunction with
the Fundamental Theorem of Calculus to justify
basic rules of integral calculus.
 Lecture 18 (November 16, 2017) (not final)
 This lecture discussed the operator norm and the
HilbertSchmidt norm of linear operators between
Euclidean spaces.
 Lecture 19 (November 20, 2017) (not final)
 This lecture discussed the definition and some
basic properties of differentiable functions of
several real variables.
 Lecture 20 (November 23, 2017) (not final)
 This lecture began by revisiting the definition
of differentiability and developed further the
basic theory of differentiable functions of
several real variables.
 Lecture 21 (November 23, 2017) (not final)
 This lecture contained proofs of the Product
Rule and Chain Rule for differentiation of
functions of several real variables, and discussed
counterexamples demonstrating that the mere existence
of partial derivatives is not sufficient to ensure
differentiability.
 Lecture 22 (November 27, 2017) (not final)
 This lecture discussed continuous differentiability,
and included a proof of the result that the existence
of continuous first order partial derivatives implies
differentiability.
 Lecture 23 (November 30, 2017) (not final)
 This lecture discussed the theory surrounding
second order partial derivatives
 Lecture 24 (November 30, 2017) (not final)
 This lecture covered proofs of results required
for the Inverse Function Theorem
 Lecture 25 (December 4, 2017) (not final)
 This lecture concerned the Inverse Function Theorem
 Lecture 26 (December 7, 2017) (not final)
 This lecture concerned the Implicit Function Theorem
 Lecture 27 (December 7, 2017)
 “Chalk and talk” discussion of problems on
scholarship papers.
 Additional Material (December 11 and 14, 2017) (not final)
Back to D.R. Wilkins: MA2321
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.