Department of Mathematics at UTSA - User contributions [en]

MAT4223

2020-08-05T14:10:25Z

Gelu.popescu:

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">7.1</div>

||

[[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

||

<div style="text-align: center;">7.2</div>

||

[[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

||

<div style="text-align: center;">7.3 and 7.4 </div>

||

[[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

||

<div style="text-align: center;"> 7.5 </div>

||

[[Weierstrass Example]]

||

* Uniform Convergence of Sequences of Functions

||

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

||

|-

|Week 3

||

<div style="text-align: center;">8.1 </div>

||

[[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

||

<div style="text-align: center;">8.2 </div>

||

[[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

||

<div style="text-align: center;">8.3 and 8.4</div>

||

[[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

||

<div style="text-align: center;">9.1</div>

||

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

||

* Limits of sequences of numbers

||

* Convergence theorems
* Bolzano-Weierstrass Theorem

||

|-
|Week 4

||

<div style="text-align: center;">9.2</div>

||

[[ The Heine-Borel Theorem]]

||

* Bolzano-Weierstrass Theorem

||

* Understand the structure of compact sets
* Applications
||

|-

|Week 5

||

<div style="text-align: center;">9.3 </div>

||

[[Limits of Functions]]

||

* Limits of real-valued functions

||

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

||

|-

|Week 5

||

<div style="text-align: center;"> 9.4</div>

||

[[Continuous Functions]]

||

* Continuous real-valued functions

||

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

||

|-

|Week 6

||

<div style="text-align: center;">11.1</div>

||

[[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

||

<div style="text-align: center;">11.2</div>
||

[[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

||

<div style="text-align: center;"> 11.3 and 11.4</div>

||

[[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

||

<div style="text-align: center;">11.5 </div>

||

[[Mean-Value Theorems]]

||

* Mean-value theorems for real-valued functions

||

* Mean-value theorems for vector-valued functions
* Lipschitz functions

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Taylor's Formula in several variables]]

||

Taylor's formula for real-valued functions

||

* Taylor's Formula in several variables, applications

|-

|Weeks 8 and 9

||

<div style="text-align: center;">11.6</div>

||

[[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

||

<div style="text-align: center;">9.6</div>

||

[[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

||

<div style="text-align: center;">12.1</div>

||

[[Integration in the Euclidean space: Jordan regions and Volume]]

||

* Darboux integration for real-valued functions

||

* Jordan regions: volume and properties

|-

|Weeks 13 and  14

||

<div style="text-align: center;">12.2</div>

||

[[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

||

<div style="text-align: center;">12.3</div>

||

[[Iterated Integrals and Fubini's Theorem]]

||

* Basic integration theory from the previous section

||

* Fubini's theorem and applications

|-

MAT4223

2020-08-05T13:50:00Z

Gelu.popescu:

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">7.1</div>

||

[[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

||

<div style="text-align: center;">7.2</div>

||

[[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

||

<div style="text-align: center;">7.3 and 7.4 </div>

||

[[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

||

<div style="text-align: center;"> 7.5 </div>

||

[[Weierstrass Example]]

||

* Uniform Convergence of Sequences of Functions

||

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

||

|-

|Week 3

||

<div style="text-align: center;">8.1 </div>

||

[[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

||

<div style="text-align: center;">8.2 </div>

||

[[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

||

<div style="text-align: center;">8.3 and 8.4</div>

||

[[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

||

<div style="text-align: center;">9.1</div>

||

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

||

* Limits of sequences of numbers

||

* Convergence theorems
* Bolzano-Weierstrass Theorem

||

|-
|Week 4

||

<div style="text-align: center;">9.2</div>

||

[[ The Heine-Borel Theorem]]

||

* Bolzano-Weierstrass Theorem

||

* Understand the structure of compact sets
* Applications
||

|-

|Week 5

||

<div style="text-align: center;">9.3 </div>

||

[[Limits of Functions]]

||

* Limits of real-valued functions

||

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

||

|-

|Week 5

||

<div style="text-align: center;"> 9.4</div>

||

[[Continuous Functions]]

||

* Continuous real-valued functions

||

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

||

|-

|Week 6

||

<div style="text-align: center;">11.1</div>

||

[[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

||

<div style="text-align: center;">11.2</div>
||

[[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

||

<div style="text-align: center;"> 11.3 and 11.4</div>

||

[[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

||

<div style="text-align: center;">11.5 </div>

||

[[Mean-Value Theorems]]

||

* Mean-value theorems for real-valued functions

||

* Mean-value theorems for vector-valued functions
* Lipschitz functions

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Taylor's Formula in several variables]]

||

Taylor's formula for real-valued functions

||

* Taylor's Formula in several variables, applications

|-

|Weeks 8 and 9

||

<div style="text-align: center;">11.6</div>

||

[[The Inverse Function Theorem]]

||

* The inverse function theorem for real-valued functions

||

* The inverse function theorem for vector-valued functions and applications

|-

|Week 10

||

<div style="text-align: center;">9.6</div>

||

[[Lebesque theorem for Riemann Integrability on the Real Line]]

||

*

||

*

|-

|Weeks 11 and 12

||

<div style="text-align: center;">12.1</div>

||

[[Integration in Higher Dimensions: Jordan regions and Volume]]

||

*

||

*

|-

|Weeks 13 and  14

||

<div style="text-align: center;">12.2</div>

||

[[Riemann Integration on Jordan Regions in Higher Dimensions]]

||

*

||

*

|-

|Week 15

||

<div style="text-align: center;">12.3</div>

||

[[Iterated Integrals and Fubini's Theorem]]

||

*

||

*

|-

MAT4223

2020-08-05T13:41:40Z

Gelu.popescu:

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">7.1</div>

||

[[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

||

<div style="text-align: center;">7.2</div>

||

[[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

||

<div style="text-align: center;">7.3 and 7.4 </div>

||

[[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

||

<div style="text-align: center;"> 7.5 </div>

||

[[Weierstrass Example]]

||

* Uniform Convergence of Sequences of Functions

||

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

||

|-

|Week 3

||

<div style="text-align: center;">8.1 </div>

||

[[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

||

<div style="text-align: center;">8.2 </div>

||

[[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

||

<div style="text-align: center;">8.3 and 8.4</div>

||

[[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

||

<div style="text-align: center;">9.1</div>

||

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

||

* Limits of sequences of numbers

||

* Convergence theorems
* Bolzano-Weierstrass Theorem

||

|-
|Week 4

||

<div style="text-align: center;">9.2</div>

||

[[ The Heine-Borel Theorem]]

||

* Bolzano-Weierstrass Theorem

||

* Understand the structure of compact sets
* Applications
||

|-

|Week 5

||

<div style="text-align: center;">9.3 </div>

||

[[Limits of Functions]]

||

* Limits of real-valued functions

||

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

||

|-

|Week 5

||

<div style="text-align: center;"> 9.4</div>

||

[[Continuous Functions]]

||

* Continuous real-valued functions

||

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

||

|-

|Week 6

||

<div style="text-align: center;">11.1</div>

||

[[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

||

<div style="text-align: center;">11.2</div>
||

[[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

||

<div style="text-align: center;"> 11.3 and 11.4</div>

||

[[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

||

<div style="text-align: center;">11.5 </div>

||

[[Mean-Value Theorems]]

||

*

||

*

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Taylor's Formula in Higher Dimensions]]

||

*

||

*

|-

|Weeks 8 and 9

||

<div style="text-align: center;">11.6</div>

||

[[The Inverse Function Theorem]]

||

*

||

*

|-

|Week 10

||

<div style="text-align: center;">9.6</div>

||

[[Lebesque theorem for Riemann Integrability on the Real Line]]

||

*

||

*

|-

|Weeks 11 and 12

||

<div style="text-align: center;">12.1</div>

||

[[Integration in Higher Dimensions: Jordan regions and Volume]]

||

*

||

*

|-

|Weeks 13 and  14

||

<div style="text-align: center;">12.2</div>

||

[[Riemann Integration on Jordan Regions in Higher Dimensions]]

||

*

||

*

|-

|Week 15

||

<div style="text-align: center;">12.3</div>

||

[[Iterated Integrals and Fubini's Theorem]]

||

*

||

*

|-

MAT4223

2020-08-05T13:33:55Z

Gelu.popescu:

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">7.1</div>

||

[[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

||

<div style="text-align: center;">7.2</div>

||

[[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

||

<div style="text-align: center;">7.3 and 7.4 </div>

||

[[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

||

<div style="text-align: center;"> 7.5 </div>

||

[[Weierstrass Example]]

||

* Uniform Convergence of Sequences of Functions

||

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

||

|-

|Week 3

||

<div style="text-align: center;">8.1 </div>

||

[[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

||

<div style="text-align: center;">8.2 </div>

||

[[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

||

<div style="text-align: center;">8.3 and 8.4</div>

||

[[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

||

<div style="text-align: center;">9.1</div>

||

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

||

* Limits of sequences of numbers

||

* Convergence theorems
* Bolzano-Weierstrass Theorem

||

|-
|Week 4

||

<div style="text-align: center;">9.2</div>

||

[[ The Heine-Borel Theorem]]

||

* Bolzano-Weierstrass Theorem

||

* Understand the structure of compact sets
* Applications
||

|-

|Week 5

||

<div style="text-align: center;">9.3 </div>

||

[[Limits of Functions]]

||

* Limits of real-valued functions

||

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

||

|-

|Week 5

||

<div style="text-align: center;"> 9.4</div>

||

[[Continuous Functions]]

||

* Continuous real-valued functions

||

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

||

|-

|Week 6

||

<div style="text-align: center;">11.1</div>

||

[[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

||

<div style="text-align: center;">11.2</div>
||

[[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

||

<div style="text-align: center;">11.2, 11.3 and 11.4</div>

||

[[Rules for Differentiation ]]

||

*

||

*

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Mean-Value Theorems]]

||

*

||

*

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Taylor's Formula in Higher Dimensions]]

||

*

||

*

|-

|Weeks 8 and 9

||

<div style="text-align: center;">11.6</div>

||

[[The Inverse Function Theorem]]

||

*

||

*

|-

|Week 10

||

<div style="text-align: center;">9.6</div>

||

[[Lebesque theorem for Riemann Integrability on the Real Line]]

||

*

||

*

|-

|Weeks 11 and 12

||

<div style="text-align: center;">12.1</div>

||

[[Integration in Higher Dimensions: Jordan regions and Volume]]

||

*

||

*

|-

|Weeks 13 and  14

||

<div style="text-align: center;">12.2</div>

||

[[Riemann Integration on Jordan Regions in Higher Dimensions]]

||

*

||

*

|-

|Week 15

||

<div style="text-align: center;">12.3</div>

||

[[Iterated Integrals and Fubini's Theorem]]

||

*

||

*

|-

MAT4223

2020-08-05T13:00:14Z

Gelu.popescu:

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">7.1</div>

||

[[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

||

<div style="text-align: center;">7.2</div>

||

[[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

||

<div style="text-align: center;">7.3 and 7.4 </div>

||

[[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

||

<div style="text-align: center;"> 7.5 </div>

||

[[Weierstrass Example]]

||

* Uniform Convergence of Sequences of Functions

||

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

||

|-

|Week 3

||

<div style="text-align: center;">8.1 </div>

||

[[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

||

<div style="text-align: center;">8.2 </div>

||

[[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

||

<div style="text-align: center;">8.3 and 8.4</div>

||

[[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

||

<div style="text-align: center;">9.1</div>

||

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

||

* Limits of sequences of numbers

||

* Convergence theorems
* Bolzano-Weierstrass Theorem

||

|-
|Week 4

||

<div style="text-align: center;">9.2</div>

||

[[ The Heine-Borel Theorem]]

||

* Bolzano-Weierstrass Theorem

||

* Understand the structure of compact sets

||

|-

|Week 5

||

<div style="text-align: center;">9.3 and 9.4</div>

||

[[Limits of Functions]]

||

*

||

*

||

|-

|Week 5

||

<div style="text-align: center;">9.3 and 9.4</div>

||

[[Continuous Functions]]

||

*

||

*

||

|-

|Week 6

||

<div style="text-align: center;">11.1</div>

||

[[Partial Derivatives and Integrals]]

||

*

||

*

||

|-

|Week 7

||

<div style="text-align: center;">11.2, 11.3 and 11.4</div>
||

[[Differentiation in Higher Dimensions]]

||

*

||

*

||

|-

|Week 7

||

<div style="text-align: center;">11.2, 11.3 and 11.4</div>

||

[[Rules for Differentiation in Higher Dimensions]]

||

*

||

*

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Mean-Value Theorems]]

||

*

||

*

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Taylor's Formula in Higher Dimensions]]

||

*

||

*

|-

|Weeks 8 and 9

||

<div style="text-align: center;">11.6</div>

||

[[The Inverse Function Theorem]]

||

*

||

*

|-

|Week 10

||

<div style="text-align: center;">9.6</div>

||

[[Lebesque theorem for Riemann Integrability on the Real Line]]

||

*

||

*

|-

|Weeks 11 and 12

||

<div style="text-align: center;">12.1</div>

||

[[Integration in Higher Dimensions: Jordan regions and Volume]]

||

*

||

*

|-

|Weeks 13 and  14

||

<div style="text-align: center;">12.2</div>

||

[[Riemann Integration on Jordan Regions in Higher Dimensions]]

||

*

||

*

|-

|Week 15

||

<div style="text-align: center;">12.3</div>

||

[[Iterated Integrals and Fubini's Theorem]]

||

*

||

*

|-

MAT4223

2020-08-05T12:58:50Z

Gelu.popescu:

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">7.1</div>

||

[[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

||

<div style="text-align: center;">7.2</div>

||

[[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

||

<div style="text-align: center;">7.3 and 7.4 </div>

||

[[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

||

<div style="text-align: center;"> 7.5 </div>

||

[[Weierstrass Example]]

||

* Uniform Convergence of Sequences of Functions

||

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

||

|-

|Week 3

||

<div style="text-align: center;">8.1 </div>

||

[[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

||

<div style="text-align: center;">8.2 </div>

||

[[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

||

<div style="text-align: center;">8.3 and 8.4</div>

||

[[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

||

<div style="text-align: center;">9.1</div>

||

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

||

* Limits of sequences of numbers

||

* Convergence theorems
* Bolzano-Weierstrass Theorem

||

|-
|Week 4

||

<div style="text-align: center;">9.2</div>

||

[[ The Heine-Borel Theorem]]

||

* Bolzano-Weierstrass Theorem

||

* Understand the structure of compact sets

||

|-

|Week 5

||

<div style="text-align: center;">9.3 and 9.4</div>

||

[[Limits of Functions]]

||

*

||

*

||

|-

|Week 5

||

<div style="text-align: center;">9.3 and 9.4</div>

||

[[Continuous Functions]]

||

*

||

*

||

|-

|Week 6

||

<div style="text-align: center;">11.1</div>

||

[[Partial Derivatives and Integrals]]

||

*

||

*

||

|-

|Week 7

||

<div style="text-align: center;">11.2, 11.3 and 11.4</div>
||

[[Differentiation in Higher Dimensions]]

||

*

||

*

||

|-

|Week 7

||

<div style="text-align: center;">11.2, 11.3 and 11.4</div>

||

[[Rules for Differentiation in Higher Dimensions]]

||

*

||

*

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Mean-Value Theorems]]

||

*

||

*

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Taylor's Formula in Higher Dimensions]]

||

*

||

*

|-

|Weeks 8 and 9

||

<div style="text-align: center;">11.6</div>

||

[[The Inverse Function Theorem]]

||

*

||

*

|-

|Week 10

||

<div style="text-align: center;">9.6</div>

||

[[Lebesque theorem for Riemann Integrability on the Real Line]]

||

*

||

*

|-

|Weeks 11 and 12

||

<div style="text-align: center;">12.1</div>

||

[[Integration in Higher Dimensions: Jordan regions and Volume]]

||

*

||

*

|-

|Weeks 13 and  14

||

<div style="text-align: center;">12.2</div>

||

[[Riemann Integration on Jordan Regions in Higher Dimensions]]

||

*

||

*

|-

|Week 15

||

<div style="text-align: center;">12.3</div>

||

[[Iterated Integrals and Fubini's Theorem]]

||

*

||

*

|-

MAT4223

2020-08-05T12:40:54Z

Gelu.popescu:

MAT4223

2020-08-05T12:36:14Z

Gelu.popescu:

MAT4223

2020-08-05T12:31:25Z

Gelu.popescu:

MAT4223

2020-08-05T12:20:50Z

Gelu.popescu:

MAT4223

2020-08-05T12:14:51Z

Gelu.popescu:

MAT4223

2020-08-05T12:09:42Z

Gelu.popescu:

MAT4223

2020-08-05T12:07:42Z

Gelu.popescu:

MAT4223

2020-08-05T12:00:16Z

Gelu.popescu:

MAT4223

2020-08-05T11:57:01Z

Gelu.popescu:

MAT4223

2020-08-05T11:52:33Z

Gelu.popescu:

==Topics List==
{| class="wikitable sortable"
! Date !! Sections !! Topics !! Prerequisite Skills !! Student Learning Outcomes

|-

|Week 1

||

<div style="text-align: center;">7.1</div>

||

[[Uniform Convergence of Sequences of Functions]]

||

Limits of functions, continuity, differentiability, integrability

||

*

|-

|Week 1

||

<div style="text-align: center;">7.2</div>

||

[[Uniform Convergence of Series of Functions]]

||

*

||

*
|-

|Week 2

||

<div style="text-align: center;">7.3, 7.4 and 7.5</div>

||

[[Power Series and Analytic Functions]]

||

*
||

*

|-

|Week 2

||

<div style="text-align: center;">edit</div>

||

[[Weierstrass Example]]

||

*

||

*

||

|-

|Week 3

||

<div style="text-align: center;">8.1 and 8.3</div>

||

[[Euclidean Spaces: Algebraic Structure and Inner Product]]

||

*

||

*

||

|-

|Week 3

||

<div style="text-align: center;">8.2 and 8.3</div>

||

[[Linear Tranformations]]

||

*

||

*

||

|-

|Week 3

||

<div style="text-align: center;">8.3 and 8.4</div>

||

[[The Topology of Higher Dimensions: interior, closure and boundary]]

||

*

||

*

||

|-

|Week 4

||

<div style="text-align: center;">9.1 and 9.2</div>

||

[[Limits of Sequences in Higher Dimensions]]

||

*

||

*

||

|-
|Week 4

||

<div style="text-align: center;">8.3 and 8.4</div>

||

[[The Bolzano-Weierstrass Theorem and The Heine-Borel Theorem]]

||

*

||

*

||

|-

|Week 5

||

<div style="text-align: center;">9.3 and 9.4</div>

||

[[Limits of Functions]]

||

*

||

*

||

|-

|Week 5

||

<div style="text-align: center;">9.3 and 9.4</div>

||

[[Continuous Functions]]

||

*

||

*

||

|-

|Week 6

||

<div style="text-align: center;">11.1</div>

||

[[Partial Derivatives and Integrals]]

||

*

||

*

||

|-

|Week 7

||

<div style="text-align: center;">11.2, 11.3 and 11.4</div>
||

[[Differentiation in Higher Dimensions]]

||

*

||

*

||

|-

|Week 7

||

<div style="text-align: center;">11.2, 11.3 and 11.4</div>

||

[[Rules for Differentiation in Higher Dimensions]]

||

*

||

*

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Mean-Value Theorems]]

||

*

||

*

|-

|Week 8

||

<div style="text-align: center;">11.5 and 11.6</div>

||

[[Taylor's Formula in Higher Dimensions]]

||

*

||

*

|-

|Weeks 8 and 9

||

<div style="text-align: center;">11.6</div>

||

[[The Inverse Function Theorem]]

||

*

||

*

|-

|Week 10

||

<div style="text-align: center;">9.6</div>

||

[[Lebesque theorem for Riemann Integrability on the Real Line]]

||

*

||

*

|-

|Weeks 11 and 12

||

<div style="text-align: center;">12.1</div>

||

[[Integration in Higher Dimensions: Jordan regions and Volume]]

||

*

||

*

|-

|Weeks 13 and  14

||

<div style="text-align: center;">12.2</div>

||

[[Riemann Integration on Jordan Regions in Higher Dimensions]]

||

*

||

*

|-

|Week 15

||

<div style="text-align: center;">12.3</div>

||

[[Iterated Integrals and Fubini's Theorem]]

||

*

||

*

|-