MAT4223 - Revision history

Khanh at 22:37, 29 January 2022

2022-01-29T22:37:09Z

Lila: /* Topics List */

2021-11-09T20:53:43Z

Topics List

Lila: /* Topics List */

2021-10-28T23:11:28Z

Topics List

Lila: /* Topics List */

2021-10-28T23:10:05Z

Topics List

Lila at 20:17, 27 October 2021

2021-10-27T20:17:19Z

James.kercheville: Completed updated list of prerequisites

2020-08-06T00:53:37Z

Completed updated list of prerequisites

James.kercheville: Added content for the prerequisites (9.1-11.1)

2020-08-05T21:53:53Z

Added content for the prerequisites (9.1-11.1)

James.kercheville: Edited prerequisites (weeks1-3 )

2020-08-05T19:39:28Z

Edited prerequisites (weeks1-3 )

Gelu.popescu at 14:10, 5 August 2020

2020-08-05T14:10:25Z

Gelu.popescu at 13:50, 5 August 2020

2020-08-05T13:50:00Z

@@ Line 141: / Line 141: @@
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-[[Linear Tranformations]]
+[[Linear Transformations]]
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@@ Line 332: / Line 332: @@
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-[[Differentiation of Vector-valued Functions]]
+[[Derivatives of Vector Functions]]
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@@ Line 392: / Line 392: @@
 * [[The Mean Value Theorem ]]  <!-- 4213-6.1 -->
 * [[Partial Derivatives and Integrals]]  <!-- 4223-11.1 -->
-* [[Differentiation of Vector-valued Functions]]  <!-- 4223-11.2 -->
+* [[Derivatives of Vector Functions]]  <!-- 4223-11.2 -->
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@@ Line 88: / Line 88: @@
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-[[Uniform Convergence of Sequences of Functions|Weierstrass Example]]
+[[Uniform Convergence of Series of Functions|Weierstrass Example]]
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@@ Line 88: / Line 88: @@
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-[[Weierstrass Example]]
+[[Uniform Convergence of Sequences of Functions|Weierstrass Example]]
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@@ Line 223: / Line 223: @@
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-[[ The Heine-Borel Theorem]]
+[[Heine-Borel Theorem]]
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@@ Line 64: / Line 64: @@
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 * [[Power Series and Functions ]] <!-- 1224-6.1 -->
 * [[Ratio and Root Tests ]]  <!-- 1224-5.6 -->
 * [[Comparison Tests ]]  <!-- 1224-5.4 -->
 * [[Uniform Convergence of Series of Functions]] <!-- 4223-7.2 -->
@@ Line 160: / Line 160: @@
 |-
@@ Line 223: / Line 221: @@
 <div style="text-align: center;">9.2</div>
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 [[ The Heine-Borel Theorem]]
@@ Line 333: / Line 330: @@
 <div style="text-align: center;">11.2</div>
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 [[Differentiation of Vector-valued Functions]]
@@ Line 349: / Line 345: @@
 * The total derivative  of a differentiable function
 * Determine whether a function if differentiable or not
@@ Line 365: / Line 359: @@
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-[[Rules for Differentiation  and tangent planes]]
+[[Rules for Differentiation and Tangent Planes]]
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-* Rules for Differentiation of real-valued functions
+* [[Differentiation of Vector-valued Functions]]  <!-- 4223-11.2 -->
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@@ Line 390: / Line 386: @@
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-[[Mean-Value Theorems]]
+[[Mean-Value Theorems for Vector Valued Functions]]
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-* Mean-value theorems for real-valued functions
+* [[The Mean Value Theorem ]]  <!-- 4213-6.1 -->
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@@ Line 413: / Line 411: @@
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-[[Taylor's Formula in several variables]]
+[[Taylor's Formula in Several Variables]]
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-Taylor's formula for real-valued functions
+* [[Taylor's Theorem ]]  <!-- 4213-6.3 -->
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@@ Line 439: / Line 439: @@
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-−
-* The inverse function theorem for real-valued functions
+* [[Inverse Functions  ]] <!-- 4213-5.6 -->
-* Cramer's rule for solving linear systems of equations
+* [[The Geometric Interpretation of the Determinant|Cramer's Rule  ]] <!-- 2233-6.3 -->
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@@ Line 460: / Line 461: @@
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-[[Lebesque theorem for Riemann Integrability on the Real Line]]
+[[Lebesque Theorem for Riemann Integrability on the Real Line]]
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-* Riemann integrability on real-valued functions
+* [[Continuous Functions]]  <!-- 4213-5.1 -->
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 * The Riemann integrable functions are those almost everywhere continuous
@@ Line 482: / Line 485: @@
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-[[Integration in the Euclidean space: Jordan regions and Volume]]
+[[Integration in the Euclidean space: Jordan Regions and Volume]]
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-* Darboux integration for real-valued functions
+* [[Riemann Integrable Functions ]] <!-- 4213-7.2 -->
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 * Jordan regions: volume and properties
@@ Line 508: / Line 513: @@
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-* Riemann and Darboux integration for real-valued functions
+* [[Integration in the Euclidean space: Jordan Regions and Volume]] <!-- 4223-12.1 -->
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 * Definition of a Riemann integrable function, characterization, properties
-* Any continuous function on a closed Jordan region  is  integrable
+* Any continuous function on a closed Jordan region is integrable
 * Mean-valued theorems for multiple integrals
@@ Line 531: / Line 536: @@
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-* Basic integration theory from the previous section
+* [[Riemann Integration on Jordan Regions in Higher Dimensions]] <!-- 4223-12.2 -->
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@@ Line 200: / Line 200: @@
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-[[Limits of Sequences in the Euclidean space and Bolzano-Weierstrass Theorem]]
+[[Limits of Sequences in the Euclidean space and the Bolzano-Weierstrass Theorem]]
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-* Limits of sequences of numbers
+* [[Sequences and Their Limits]] <!-- 3213-3.1 -->
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-*  Convergence theorems
+* Convergence theorems
 * Bolzano-Weierstrass Theorem
@@ Line 228: / Line 230: @@
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-* Bolzano-Weierstrass Theorem
+* [[Limits of Sequences in the Euclidean space and the Bolzano-Weierstrass Theorem|The Bolzano-Weierstrass Theorem]] <!-- 4223-9.1 -->
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@@ Line 234: / Line 237: @@
 * Understand the structure of compact sets
 * Applications
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@@ Line 248: / Line 252: @@
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-[[Limits of Functions]]
+[[Limits of Vector Functions]]
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-* Limits of real-valued functions
+* [[The Definition of the Limit of a Function]]  <!-- 3213-4.1 -->
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@@ Line 274: / Line 280: @@
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-[[Continuous Functions]]
+[[Continuous Vector Functions]]
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-* Continuous real-valued functions
+* [[Continuous Functions]]  <!-- 4213-5.1 -->
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@@ Line 304: / Line 311: @@
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-* Basic results from Real Analysis I
+* [[The Derivative ]]  <!-- 4213-6.1 -->
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@@ Line 323: / Line 332: @@
 <div style="text-align: center;">11.2</div>
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-[[Differentiation of vector-valued functions]]
+[[Differentiation of Vector-valued Functions]]
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-* Differentiation of real-valued functions and geometric interpretation
+* [[Continuous Vector Functions]]  <!-- 4223-9.4 -->
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@@ Line 17: / Line 17: @@
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-*Convergence of sequences of numbers, Cauchy criterion
+* [[Sequences and Their Limits]] <!-- 3213-3.1 -->
-* Limits of functions, continuity, differentiability, Riemann integrability
+* [[The Cauchy Criterion for Convergence]] <!-- 3213-3.5 -->
+* [[The Definition of the Limit of a Function]] <!-- 3213-4.1 -->
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 *  Understand the difference between pointwise convergence and uniform convergence
 *  Understand how the continuity, diferrentiablity and integrability behave under uniform convergence
 |-
@@ Line 41: / Line 43: @@
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-* Convergence of series of numbers
+* [[Introduction to Infinite Series ]] <!-- 3213-3.7 -->
+* [[Uniform Convergence of Sequences of Functions]]  <!-- 4223-7.1 -->
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@@ Line 63: / Line 65: @@
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-* Tests for convergence of series of numbers
-* Taylor expansion formula
+* [[Power Series and Functions ]] <!-- 1224-6.1 -->
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-*  Determine the set of convergence for a power series
+* 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
@@ Line 87: / Line 92: @@
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-* Uniform Convergence of Sequences of Functions
+* [[Uniform Convergence of Series of Functions|The Weierstrass M-test]] <!-- 4223-7.1 -->
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@@ Line 110: / Line 115: @@
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-−
-* Vectors and operations with vectors, the dot product.
+* [[Vectors, Matrices, and Gauss-Jordan Elimination|Vectors and the Dot Product]] <!-- 2233- 1.2 and 1.3 -->
-* Basic knowledge of trigonometry
+* [[Properties of the Trigonometric Functions]] <!-- 1093-2.3 -->
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-* Norms  in the Euclidean space, comparison
+* Norms in the Euclidean space, comparison
 * Cauchy-Schwartz inequality, triangle inequality
@@ Line 140: / Line 145: @@
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-* Basic linear algebra concepts: matrices, linear transformations
+* [[Introduction to Linear Transformations ]] <!-- 2233-2.1 -->
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@@ Line 170: / Line 177: @@
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-* Basic notions of topology on the real line
+* [[Intervals|Closed and open intervals ]] <!-- 3213-3.7 -->
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@@ Line 415: / Line 415: @@
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-[[The Inverse Function Theorem]]
+[[The Inverse Function Theorem and the Implicit Function Theorem]]
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 * The inverse function theorem for real-valued functions
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-* The inverse function theorem for vector-valued functions and applications
+* The inverse function theorem and the implicit function theorem for vector-valued functions
@@ Line 441: / Line 443: @@
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+* Riemann integrability on real-valued functions
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+* The Riemann integrable functions are those almost everywhere continuous
@@ Line 459: / Line 461: @@
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-[[Integration in Higher Dimensions: Jordan regions and Volume]]
+[[Integration in the Euclidean space: Jordan regions and Volume]]
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+* Darboux integration for real-valued functions
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+* Jordan regions: volume and properties
@@ Line 485: / Line 487: @@
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+* Riemann and Darboux integration for real-valued functions
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+* Definition of a Riemann integrable function, characterization, properties
+* Any continuous function on a closed Jordan region  is  integrable
 |-
@@ Line 507: / Line 510: @@
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+* Basic integration theory from the previous section
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+* Fubini's theorem and applications
 |-

@@ Line 373: / Line 373: @@
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+* Mean-value theorems for real-valued functions
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+* Mean-value theorems for vector-valued functions
@@ Line 391: / Line 392: @@
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-[[Taylor's Formula in Higher Dimensions]]
+[[Taylor's Formula in several variables]]
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+Taylor's formula for real-valued functions
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+* Taylor's Formula in several variables, applications
@@ Line 418: / Line 419: @@
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+* The inverse function theorem for real-valued functions
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+* The inverse function theorem for vector-valued functions and applications