MAT4223

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	7.1	Uniform Convergence of Sequences of Functions	Sequences and Their Limits The Cauchy Criterion for Convergence The Definition of the Limit of a Function Continuous Functions on Intervals The Derivative Riemann Integrable Functions	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	Introduction to Infinite Series Uniform Convergence of Sequences of Functions	Learn the term-by-term differentiation (resp. integration) theorem Weierstrass M-test
Week 2	7.3 and 7.4	Power Series and Analytic Functions	Power Series and Functions Ratio and Root Tests Comparison Tests Taylor's Theorem Uniform Convergence of Series of Functions	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	The Weierstrass M-test	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 the Dot Product Properties of the Trigonometric Functions	Norms in the Euclidean space, comparison Cauchy-Schwartz inequality, triangle inequality
Week 3	8.2	Linear Transformations	Introduction to Linear Transformations Basics of Matrices Euclidean Spaces: Algebraic Structure and Inner Product	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	Closed and open intervals Cluster Points Absolute Value and the Real Line	Learn basic notions of topology in the Euclidean space
Week 4	9.1	Limits of Sequences in the Euclidean space and the Bolzano-Weierstrass Theorem	Sequences and Their Limits The Cauchy Criterion for Convergence Subsequences	Convergence theorems Bolzano-Weierstrass Theorem
Week 4	9.2	Heine-Borel Theorem	The Bolzano-Weierstrass Theorem The Topology of Higher Dimensions: interior, closure and boundary	Understand the structure of compact sets Applications
Week 5	9.3	Limits of Vector Functions	The Definition of the Limit of a Function The Limit Theorems for Functions The Topology of Higher Dimensions: interior, closure and boundary	Limit theorems, sequential criterion Iterated limits Commutation of iterated limits
Week 5	9.4	Continuous Vector Functions	Continuous Functions The Topology of Higher Dimensions: interior, closure and boundary	Characterizations of continuity Properties of continuous functions Extreme value theorem
Week 6	11.1	Partial Derivatives and Integrals	The Derivative The Riemann Integral Continuous Functions	Commutation of partial derivatives Commutation of the limit with the the integral Differentiating under the integral sign
Week 7	11.2	Derivatives of Vector Functions	Continuous Vector Functions Partial Derivatives and Integrals Euclidean Spaces: Algebraic Structure and Inner Product	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	Differentiation of Vector-valued Functions The Derivative Euclidean Spaces: Algebraic Structure and Inner Product	Basic rules for differentiation and applications Tangent hyperplanes to surfaces Chain rule
Week 8	11.5	Mean-Value Theorems for Vector Valued Functions	The Mean Value Theorem Partial Derivatives and Integrals Derivatives of Vector 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 Theorem Power Series and Analytic Functions Mean-Value Theorems for Vector Valued Functions	Taylor's Formula in several variables, applications
Weeks 8 and 9	11.6	The Inverse Function Theorem and the Implicit Function Theorem	Inverse Functions Cramer's Rule Mean-Value Theorems for Vector Valued Functions	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	Continuous Functions Riemann Integrable Functions	Define an almost everywhere continuous function The Riemann integrable functions are those almost everywhere continuous
Weeks 11 and 12	12.1		Riemann Integrable Functions The Darboux Integral	Jordan regions: volume and properties
Weeks 13 and  14	12.2	Riemann Integration on Jordan Regions in Higher Dimensions	Integration in the Euclidean space: Jordan Regions and Volume	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	Riemann Integration on Jordan Regions in Higher Dimensions	Fubini's theorem and applications