Course Overview
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Course Synopsis
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This course gives students understanding and knowledge of complex numbers together with their derivatives, manipulation, and other properties of differential geometry of curves and surfaces.
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Course Learning Outcomes
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Upon completing this course students should be able to:
- demonstrate understanding of the basic concepts, results and geometric interpretation of complex analysis.
- identify the singular points of a complex function.
- explain the residue concept and general Cauchy integral theorem and formula.
- calculate the regular curves, arc length, curvature, torsion of a curve, and parametrization.
- find the osculating surface and the osculating curve at any point of a given curve.
- calculate the first and the second fundamental forms of a surface, the Gaussian , the mean, and geodesic curvatures.
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Course Calendar
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Week 01
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2
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Representation of Complex Numbers
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3
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The Algebra of Complex Numbers
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4
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Addition of Complex Numbers
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5
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Multiplication of Complex Numbers
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6
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The Geometry of Complex Numbers (I)
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7
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The Geometry of Complex Numbers (II)
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11
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Product and Powers in Exponential Form
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Week 02
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12
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Roots of Complex Numbers (I)
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13
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Roots of Complex Numbers (II)
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14
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Regions in Complex Plane (I)
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15
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Regions in Complex Plane (II)
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16
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Regions in Complex Plane (III)
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18
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Complex Functions (II)
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Week 03
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20
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Mappings: Linear Transformation (I)
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21
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Mappings: Linear Transformation (II)
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22
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Geometry of Mappings (I)
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23
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Geometry of Mappings (II)
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24
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Geometry of Mappings (III)
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Assignment No. 1
Week 04
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28
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Theorems on Limits (I)
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32
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Consequences of Continuity
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33
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The Reciprocal Transformation
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Week 05
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37
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The Cauchy–Riemann Equations (I)
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38
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The Cauchy–Riemann Equations (II)
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39
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The Cauchy–Riemann Equations (III)
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40
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The Cauchy-Riemann Equation in Polar Coordinates
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Week 06
Quiz No. 1
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43
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Analytic Functions-III
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Week 07
Quiz No. 2
Week 08
Mid Term Exam
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59
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Power Series Functions-I
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60
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Power Series Functions-II
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61
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Power Series Functions-III
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62
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Power Series Functions Term by Term Differentiation_I
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63
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Power Series Functions Term by Term Differentiation_II
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64
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Power Series Functions Term by Term Differentiation_III
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Week 09
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65
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Complex Exponential Functions_I
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66
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Complex Exponential Functions_II
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67
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Complex Exponential Functions_III
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Week 10
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75
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Complex Components-III
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Assignment No. 2
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76
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Trignometric Functions-I
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77
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Trignometric Functions-II
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78
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Trignometric Functions-III
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79
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Trignometric Functions-IV
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80
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Trignometric Functions-V
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Week 11
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82
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Inverse Trignometric Functiuons- I
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83
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Inverse Trignometric Functiuons- II
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84
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Inverse Trignometric Functiuons- III
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90
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Complex Integrals- III
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Week 12
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93
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Contour Integrals- III
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95
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Properties of Contour Integrals
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96
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Inequalities involving Contour Integrals
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97
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The Cauchy-Goursat Theorem-I
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98
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The Cauchy-Goursat Theorem-II
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99
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The Cauchy-Goursat Theorem-III
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100
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Generalization of The Cauchy-Goursat Theorem
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Quiz No. 3
Week 13
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101
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Fundamental Theorem of Integration
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102
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Cauchy's Integral Formula-I
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103
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Cauchy's Integral Formula-II
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104
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Cauchy's Integral Formula For Derivataives-I
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105
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Cauchy's Integral Formula For Derivataives-II
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106
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Maximum Modulus Principle
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Week 14
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108
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Introduction to Differential Geometry
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111
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Directional Derivatives-I
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112
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Directional Derivatives-II
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Week 15
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116
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Geometrical Properties of Curves
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117
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Arc Length Reparameterization
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118
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Vector Fields on Curves
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Final Term Exam
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