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Trinity College Dublin

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) (not final)
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) (not final)
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) (not final)
This lecture developed the theory of the Riemann integral for functions of a single real variable.
Lecture 12 (October 23, 2017) (not final)
n
This lecture developed the theory of the Riemann integral for functions of a single real variable.
Lecture 13 (October 26, 2017) (not final)
This lecture discussed partitions of closed n-dimensional cells in
Lecture 14 (October 26, 2017) (not final)
This lecture presented 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 2, 2017) (not final)
This lecture developed the theory of the Riemann integral for functions of several real variables.

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