Course Overview
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Course Synopsis
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Real Analysis II is the follow up course of Real Analysis I and in general an advanced course related to mathematical analysis. The topics of the Real Analysis II are linked with its rst course namely Real Analysis I indeed we will extend the ideas of Real Analysis I to Euclidean space Rn we will discuss sequences and series of functions limits and continuity of functions of several variables partial derivatives their applications multiple integrals etc.
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Course Learning Outcomes
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Upon completion of this course students will be able to
- Understand the principles theorems and applications related to sequences and series of functions.
- Apply knowledge of power series including Taylor series Maclaurin series and Fourier series.
- Extend the principles of real analysis to functions of multiple variables covering topics such as continuity partial derivatives chain rule and multiple integrals.
- Effectively use the acquired knowledge from Real Analysis II in advanced studies.
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Course Calendar
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Week 01
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3
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pointwise convergence: Examples
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5
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Uniform convergence theorems
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6
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Uniform convergence: Counter Examples
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7
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Uniform convergence: Theorems (Continue)
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8
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Cauchy's Theorem (Examples)
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9
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Properties of uniform convergent functions
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10
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Properties of uniform convergent functions (Continue)
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11
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Properties of uniform convergent functions (Continue..)
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Week 02
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12
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Properties of uniform convergent functions (Examples)
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13
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Properties of uniform convergent functions (Examples cont.)
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14
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Properties of uniform convergent functions (Theorems)
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15
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Infinite Series of functions (Pointwise and Uniform convergence)
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16
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Infinite Series of functions (Examples)
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17
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Infinite Series of functions (Counter examples)
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18
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Cauchy's criterion of Functional series
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19
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Cauchy's criterion of Functional series (Examples)
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20
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Dominated series on set S
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22
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Weierstrass's M-Test: Examples
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Week 03
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23
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Dirichlet's test for uniform convergence
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24
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Dirichlet's test for uniform convergence (Continue)
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25
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Series of product of two functions
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26
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Series of product of two functions: Examples
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27
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Continuity of uniformly convergent series: Examples
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28
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Integration of sequences of integrable functions
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29
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Interchange of Summation and Integration
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30
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Interchange of Summation and Integration: Examples
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31
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Interchange of Summation and Integration (Continue)
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32
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Interchange of limit and differentiation: Examples
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Week 04
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33
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Differentiation of a sequence of functions
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34
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Interchange of summation and differentiation of infinite series
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35
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Interchange of summation and differentiation of infinite series: Examples
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37
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Properties of Power Series
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38
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Properties of Power Series (Continue)
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39
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Properties of Power Series: Examples
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40
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Properties of Power Series: Theorems
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41
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Definate integral of a function represented by power series
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43
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Taylor Series of functions: Theorem
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Week 05
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44
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Taylor Series of functions: Examples
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45
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Taylor Series of functions: Examples (Continue)
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46
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Arithematic operations with power series: Theorem
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47
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Product of power serie: Theorem
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48
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Arithematic operations with power series: Examples
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49
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The reciprocal of pwer series: Example
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51
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The Abel's Theorem: Examples
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52
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Equicontinuous functions on a set
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53
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Convergent subsequences
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54
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Equicontinuous functions: Theorem
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55
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Equicontinuous functions (Continue)
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Week 06
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56
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The Stone-Weierstrass Theorem
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58
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Properties of Periodic functions
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59
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Properties of Periodic functions: Lemma
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60
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Fourier Series: Trignometric Polynomials
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61
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Fourier Series: Vector space and Inner Product: Lemma
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62
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Fourier Series: Orthogonal set of functions
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63
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Fourier Series: Basic results
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64
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Fourier Series: Even and odd functions
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65
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Fourier Series: Even and odd functions: Examples
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66
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Convergence of Fourier Series
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Week 07
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68
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Dirichlet Theorem: Examples
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69
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Dirichlet Theorem: Examples (Continue)
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70
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Fourier Series of functions with arbitrary period
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71
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Fourier Series of functions with arbitrary period: Examples
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72
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Best Approximation Theorem
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73
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Functions of several variables
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74
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Inner product and Schwarz's inequality in R^n
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75
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Applications of Schwarz's inequaity
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76
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Some properties of R^n
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77
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Line segments, Neighbourhoods and open sets in R^n
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79
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Cauhcy's Convergence Criterion
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80
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Principle of Nested Sets: Theorem
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Week 08
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81
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Heine-Borel Theorem in R^n
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83
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Polygonally connected set in R^n
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84
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Region in R^n: Examples
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86
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Functions of several variables: Definition
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87
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Functions of several variables: Examples
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89
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Infinite limits: Examples
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90
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Limits at infinity: Examples
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Week 09
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91
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Limits at infinity: Examples (Continue)
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92
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Continuity of functions of n variables
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93
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Continuity of functions of n variables: Examples
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94
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Vector-Valued Functions
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95
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Composite vector valued Functions
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96
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Composite vector valued Functions: Examples
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97
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Composite vector valued Functions: Examples (Continue)
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99
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Intermedate value Theorem
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101
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Directional Derivative
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Week 10
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102
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Partial derivatives: Examples
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103
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Partial derivatives: Examples (Continue)
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104
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Equality of mixed partial derivatives
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105
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Genralization of equality of mixed derivatives
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106
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Differentiability of Functions of Several Variables
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107
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Differentiability Rresults
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109
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The differential in functions of several variables
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110
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The differential: Examples
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111
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The differential: Lemma
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112
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Sufficient condition for differentiability: Theorem
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113
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Sufficient condition for differentiability: Examples
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114
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Geometric interpretations of differentiablity
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Week 11
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115
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Maxima and Minima for function of n variables
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116
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Maxima and Minima for functions of n variables
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117
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The chain rule in function of several variables
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118
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The Chain Rule: Examples
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119
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Higher derivatives of Composite function
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120
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Higher derivatives of Composite functions: Examples
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121
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Higher derivatives of Composite functions: Theorem
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122
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The Mean Value Theorem
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123
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Higher Differentia: Examples
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124
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Taylor’s Theorem for Functions of n Variables
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125
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Taylor’s series: Examples
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126
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Taylor’s series: Theorem
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Week 12
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127
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Definite polynomials types: Examples
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128
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Extreme Values: Theorem
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129
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Extreme Values: Examples
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130
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Improper Integrals Introduction
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131
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Locally integrable functions
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132
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Locally integrable functions: Examples
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133
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Improper intergral: Theorem
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134
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Improper Intgerals of Nonnegative functions
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136
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The Comparison test: Examples
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137
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Absolute intgerability
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138
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Absolute intgerability: Theorem
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Week 13
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139
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Conditional convergence of improper integrals
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141
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The Dirichlet's Test: Examples
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142
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Rectangles in R^n and partitions
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144
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Riemann sum: Examples
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146
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Upper and Lower Integrals
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147
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Upper and Lower Integrals for a bounde function: Theorem
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148
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Upper and Lower Integrals for a bounde function: Examples
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149
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Some properties: Theorem
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150
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Intgerals over more general subsets of R^n
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Week 14
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151
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Differentiable surfaces with examples
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153
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Properties of multiple intgerals and theorems witout proofs
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154
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Itegrated intgerals and Examples
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155
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Fubini's Theorem Examples
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156
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Some results: Theorems
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