From Department of Mathematics at UTSA
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Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 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

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

Weierstrass Example

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

Week 3

Euclidean Spaces: Algebraic Structure and Inner Product

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

Week 3

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

  • Learn basic notions of topology in the Euclidean space

Week 4

Limits of Sequences in the Euclidean space and the Bolzano-Weierstrass Theorem

  • Convergence theorems
  • Bolzano-Weierstrass Theorem

Week 4

Heine-Borel Theorem

  • Understand the structure of compact sets
  • Applications

Week 5

Limits of Vector Functions

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

Week 5

Continuous Vector Functions

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

Week 6

Partial Derivatives and Integrals

  • Commutation of partial derivatives
  • Commutation of the limit with the the integral
  • Differentiating under the integral sign

Week 7

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

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

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

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

Integration in the Euclidean space: Jordan Regions and Volume

  • Jordan regions: volume and properties

Weeks 13 and  14

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

Iterated Integrals and Fubini's Theorem

  • Fubini's theorem and applications