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

  • 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

  • Learn the term-by-term differentiation (resp. integration) theorem
  • Weierstrass M-test
Week 2
7.3 and 7.4

Power Series and Analytic 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

  • Learn to construct a continuous function everywhere which is not differentiable anywhere


Week 3
8.1

Euclidean Spaces: Algebraic Structure and Inner Product

  • Norms in the Euclidean space, comparison
  • Cauchy-Schwartz inequality, triangle inequality


Week 3
8.2

Linear Tranformations

  • 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

  • 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

  • Convergence theorems
  • Bolzano-Weierstrass Theorem


Week 4
9.2

Heine-Borel Theorem

  • Understand the structure of compact sets
  • Applications


Week 5
9.3

Limits of Vector Functions

  • Limit theorems, sequential criterion
  • Iterated limits
  • Commutation of iterated limits


Week 5
9.4

Continuous Vector Functions

  • Characterizations of continuity
  • Properties of continuous functions
  • Extreme value theorem


Week 6
11.1

Partial Derivatives and Integrals

  • 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

  • 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

  • Basic rules for differentiation and applications
  • Tangent hyperplanes to surfaces
  • Chain rule


Week 8
11.5

Mean-Value Theorems for Vector 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 in several variables, applications


Weeks 8 and 9
11.6

The Inverse Function Theorem and the Implicit Function Theorem

  • 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

  • Define an almost everywhere continuous function
  • The Riemann integrable functions are those almost everywhere continuous


Weeks 11 and 12
12.1

Integration in the Euclidean space: Jordan Regions and Volume

  • Jordan regions: volume and properties


Weeks 13 and  14
12.2

Riemann Integration on Jordan Regions in Higher Dimensions

  • 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

  • Fubini's theorem and applications