MAT4223

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	7.1	Uniform Convergence of Sequences of Functions	Convergence of sequences of numbers, Cauchy criterion Limits of functions, continuity, differentiability, Riemann integrability	Understand the difference between pointwise convergence and uniform convergence Understand how the continuity, diferrentiablity and integrability behave under uniform convergence
Week 1	7.2	Uniform Convergence of Series of Functions	Convergence of series of numbers	Learn the term-by-term differentiation (resp. integration) theorem Weierstrass M-test
Week 2	7.3 and 7.4	Power Series and Analytic Functions	Tests for convergence of series of numbers Taylor expansion formula	Determine the set of convergence for a power series Apply the results of the previous sections to power series. Apply the results to the elementary functions
Week 2	7.5	Weierstrass Example	Uniform Convergence of Sequences of Functions	Learn to construct a continuous function everywhere which is not differentiable anywhere
Week 3	8.1	Euclidean Spaces: Algebraic Structure and Inner Product	Vectors and operations with vectors, the dot product. Basic knowledge of trigonometry	Norms in the Euclidean space, comparison Cauchy-Schwartz inequality, triangle inequality
Week 3	8.2	Linear Tranformations	Basic linear algebra concepts: matrices, linear transformations	The matrix of a linear transformation The norm of a linear transformation Write equations of hyperplanes in the Euclidean space
Week 3	8.3 and 8.4	The Topology of Higher Dimensions: interior, closure and boundary	Basic notions of topology on the real line	Learn basic notions of topology in the Euclidean space
Week 4	9.1	Limits of Sequences in the Euclidean space and Bolzano-Weierstrass Theorem	Limits of sequences of numbers	Convergence theorems Bolzano-Weierstrass Theorem
Week 4	9.2	The Heine-Borel Theorem	Bolzano-Weierstrass Theorem	Understand the structure of compact sets Applications
Week 5	9.3	Limits of Functions	Limits of real-valued functions	Limit theorems, sequential criterion Iterated limits Commutation of iterated limits
Week 5	9.4	Continuous Functions	Continuous real-valued functions	Characterizations of continuity Properties of continuous functions Extreme value theorem
Week 6	11.1	Partial Derivatives and Integrals	Basic results from Real Analysis I	Commutation of partial derivatives Commutation of the limit with the the integral Differentiating under the integral sign
Week 7	11.2	Differentiation of vector-valued functions	Differentiation of real-valued functions and geometric interpretation	Necessary and sufficient conditions for differentiation The total derivative of a differentiable function Determine whether a function if differentiable or not
Week 7	11.3 and 11.4	Rules for Differentiation and tangent planes	Rules for Differentiation of real-valued functions	Basic rules for differentiation and applications Tangent hyperplanes to surfaces Chain rule
Week 8	11.5	Mean-Value Theorems	Mean-value theorems for real-valued functions	Mean-value theorems for vector-valued functions Lipschitz functions
Week 8	11.5 and 11.6	Taylor's Formula in several variables	Taylor's formula for real-valued functions	Taylor's Formula in several variables, applications
Weeks 8 and 9	11.6	The Inverse Function Theorem and the Implicit Function Theorem	The inverse function theorem for real-valued functions Cramer's rule for solving linear systems of equations	The inverse function theorem and the implicit function theorem for vector-valued functions Applications
Week 10	9.6	Lebesque theorem for Riemann Integrability on the Real Line	Riemann integrability on real-valued functions	The Riemann integrable functions are those almost everywhere continuous
Weeks 11 and 12	12.1		Darboux integration for real-valued functions	Jordan regions: volume and properties
Weeks 13 and  14	12.2	Riemann Integration on Jordan Regions in Higher Dimensions	Riemann and Darboux integration for real-valued functions	Definition of a Riemann integrable function, characterization, properties Any continuous function on a closed Jordan region is integrable Mean-valued theorems for multiple integrals
Week 15	12.3	Iterated Integrals and Fubini's Theorem	Basic integration theory from the previous section	Fubini's theorem and applications