MAT4223

From Department of Mathematics at UTSA
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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


Week 5
9.3 and 9.4

Limits of Functions


Week 5
9.3 and 9.4

Continuous Functions


Week 6
11.1

Partial Derivatives and Integrals


Week 7
11.2, 11.3 and 11.4


Differentiation in Higher Dimensions


Week 7
11.2, 11.3 and 11.4

Rules for Differentiation in Higher Dimensions


Week 8
11.5 and 11.6

Mean-Value Theorems


Week 8
11.5 and 11.6

Taylor's Formula in Higher Dimensions


Weeks 8 and 9
11.6

The Inverse Function Theorem


Week 10
9.6

Lebesque theorem for Riemann Integrability on the Real Line


Weeks 11 and 12
12.1

Integration in Higher Dimensions: Jordan regions and Volume


Weeks 13 and  14
12.2

Riemann Integration on Jordan Regions in Higher Dimensions


Week 15
12.3

Iterated Integrals and Fubini's Theorem