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 Dedekind-complete ordered field.

Lecture 2 (September 28, 2017)

This lecture covered basic results concerning convergence of infinite sequences of real numbers, including the one-dimensional case of the Bolzano-Weierstrass Theorem.

Lecture 3 (September 28, 2017)

This lecture covered basic results concerning the scalar product and Euclidean norm of vectors in ℝⁿ, including Schwarz's Inequality and the Triangle Inequality. The definition of convergence was presented for sequences of points in Euclidean spaces ℝⁿ of arbitrary (finite) dimension n. The lecture also presented the definitions of limit points and isolated points of subsets of ℝⁿ.

Lecture 4 (October 2, 2017)

This lecture covered versions of the Bolzano-Weierstrass Theorem and Cauchy's Criterion for Convergence applicable to bounded sequences of points in Euclidean spaces ℝⁿ 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 Bolzano-Weierstrass 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 finite-dimensional 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)

This lecture discussed partitions of closed n-dimensional cells in ℝⁿ

Lecture 13 (October 26, 2017)

This lecture developed the definition of the Riemann integral for functions of several real variables.

Lecture 14 (November 2, 2017)

This lecture developed the definition of the Riemann integral for functions of several real variables.

Lecture 15 (November 2, 2017)

This lecture developed the theory of the Riemann integral for functions of several real variables.

Lecture 16 (November 13, 2017)

This lecture reviewed the basic theory of differentiation, for functions of one real variable.

Lecture 17 (November 16, 2017)

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)

This lecture discussed the operator norm and the Hilbert-Schmidt norm of linear operators between Euclidean spaces.

Lecture 19 (November 20, 2017)

This lecture discussed the definition and some basic properties of differentiable functions of several real variables.

Lecture 20 (November 23, 2017)

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)

This lecture contained proofs of the Product Rule and Chain Rule for differentiation of functions of several real variables, and discussed counter-examples demonstrating that the mere existence of partial derivatives is not sufficient to ensure differentiability.

Lecture 22 (November 27, 2017)

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)

This lecture discussed the theory surrounding second order partial derivatives

Lecture 24 (November 30, 2017)

This lecture covered proofs of results required for the Inverse Function Theorem

Lecture 25 (December 4, 2017)

This lecture concerned the Inverse Function Theorem

Lecture 26 (December 7, 2017)

This lecture concerned the Implicit Function Theorem

Lecture 27 (December 7, 2017)

“Chalk and talk” discussion of problems on scholarship papers.

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Dr. David R. Wilkins, School of Mathematics, Trinity College Dublin.