| Preface |
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vii | |
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Vector Algebra I: Scalars and Vectors |
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1 | (22) |
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1 | (3) |
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4 | (2) |
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Sum of Two Vectors: Geometrical Addition |
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4 | (2) |
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6 | (1) |
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Components and Projection of a Vector |
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7 | (2) |
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Component Representation in Coordinate Systems |
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9 | (5) |
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9 | (1) |
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10 | (1) |
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Component Representation of a Vector |
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11 | (1) |
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Representation of the Sum of Two Vectors in Terms of Their Components |
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12 | (1) |
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Subtraction of Vectors in Terms of their Components |
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13 | (1) |
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Multiplication of a Vector by a Scalar |
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14 | (1) |
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15 | (8) |
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Vector Algebra II: Scalar and Vector Products |
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23 | (16) |
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23 | (7) |
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Application: Equation of a Line and a Plane |
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26 | (1) |
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26 | (1) |
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Commutative and Distributive Laws |
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27 | (1) |
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Scalar Product in Terms of the Components of the Vectors |
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27 | (3) |
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30 | (9) |
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30 | (1) |
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31 | (1) |
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Definition of the Vector Product |
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32 | (1) |
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33 | (1) |
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Anti-Commutative Law for Vector Products |
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33 | (1) |
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Components of the Vector Product |
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34 | (5) |
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39 | (30) |
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The Mathematical Concept of Functions and its Meaning in Physics and Engineering |
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39 | (3) |
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39 | (1) |
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The Concept of a Function |
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40 | (2) |
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Graphical Representation of Functions |
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42 | (5) |
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Coordinate System, Position Vector |
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42 | (1) |
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The Linear Function: The Straight Line |
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43 | (1) |
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44 | (3) |
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47 | (2) |
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Parametric Changes of Functions and Their Graphs |
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49 | (1) |
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50 | (2) |
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Trigonometric or Circular Functions |
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52 | (12) |
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52 | (1) |
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53 | (5) |
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58 | (1) |
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Relationships Between the Sine and Cosine Functions |
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59 | (2) |
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61 | (1) |
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62 | (2) |
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Inverse Trigonometric Functions |
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64 | (2) |
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Function of a Function (Composition) |
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66 | (3) |
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Exponential, Logarithmic and Hyperbolic Functions |
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69 | (16) |
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Powers, Exponential Function |
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69 | (5) |
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69 | (1) |
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Laws of Indices or Exponents |
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70 | (1) |
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71 | (1) |
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71 | (3) |
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Logarithm, Logarithmic Function |
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74 | (4) |
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74 | (2) |
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Operations with Logarithms |
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76 | (1) |
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77 | (1) |
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Hyperbolic Functions and Inverse Hyperbolic Functions |
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78 | (7) |
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78 | (3) |
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Inverse Hyperbolic Functions |
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81 | (4) |
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85 | (60) |
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85 | (6) |
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85 | (1) |
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86 | (3) |
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89 | (1) |
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Examples for the Practical Determination of Limits |
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89 | (2) |
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91 | (1) |
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92 | (2) |
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93 | (1) |
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Differentiation of a Function |
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94 | (6) |
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Gradient or Slope of a Line |
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94 | (1) |
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Gradient of an Arbitrary Curve |
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95 | (2) |
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97 | (1) |
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Physical Application: Velocity |
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98 | (1) |
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99 | (1) |
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Calculating Differential Coefficients |
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100 | (12) |
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Derivatives of Power Functions; Constant Factors |
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101 | (1) |
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Rules for Differentiation |
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102 | (4) |
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Differentiation of Fundamental Functions |
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106 | (6) |
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112 | (1) |
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Extreme Values and Points of Inflexion; Curve Sketching |
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113 | (8) |
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Maximum and Minimum Values of a Function |
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113 | (4) |
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Further Remarks on Points of Inflexion (Contraflexure) |
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117 | (1) |
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118 | (3) |
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Applications of Differential Calculus |
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121 | (6) |
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121 | (1) |
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122 | (1) |
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123 | (2) |
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Determination of Limits by Differentiation: L'Hopital's Rule |
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125 | (2) |
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Further Methods for Calculating Differential Coefficients |
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127 | (2) |
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Implicit Functions and their Derivatives |
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127 | (1) |
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Logarithmic Differentiation |
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128 | (1) |
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Parametric Functions and their Derivatives |
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129 | (16) |
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Parametric Form of an Equation |
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129 | (4) |
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Derivatives of Parametric Functions |
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133 | (12) |
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145 | (46) |
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145 | (2) |
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Fundamental Problem of Integral Calculus |
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145 | (2) |
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The Area Problem: The Definite Integral |
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147 | (2) |
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Fundamental Theorem of the Differential and Integral Calculus |
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149 | (4) |
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153 | (6) |
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Calculation of Definite Integrals from Indefinite Integrals |
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153 | (3) |
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Examples of Definite Integrals |
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156 | (3) |
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159 | (16) |
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Principle of Verification |
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159 | (1) |
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159 | (1) |
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Constant Factor and the Sum of Functions |
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160 | (1) |
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Integration by Parts: Product of Two Functions |
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161 | (3) |
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Integration by Substitution |
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164 | (2) |
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Substitution in Particular Cases |
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166 | (4) |
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Integration by Partial Fractions |
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170 | (5) |
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Rules for Solving Definite Integrals |
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175 | (3) |
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178 | (1) |
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179 | (2) |
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181 | (10) |
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Applications of Integration |
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191 | (36) |
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191 | (7) |
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Areas for Parametric Functions |
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194 | (1) |
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Areas in Polar Coordinates |
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195 | (2) |
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197 | (1) |
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198 | (4) |
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Lengths of Curves in Polar Coordinates |
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201 | (1) |
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Surface Area and Volume of a Solid of Revolution |
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202 | (6) |
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Applications to Mechanics |
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208 | (19) |
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Basic Concepts of Mechanics |
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208 | (1) |
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Center of Mass and Centroid |
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208 | (3) |
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211 | (2) |
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Moments of Inertia; Second Moment of Area |
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213 | (14) |
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Taylor Series and Power Series |
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227 | (20) |
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227 | (1) |
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Expansion of a Function in a Power Series |
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228 | (4) |
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Interval of Convergence of Power Series |
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232 | (1) |
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Approximate Values of Functions |
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233 | (2) |
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Expansion of a Function f (x) at an Arbitrary Position |
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235 | (2) |
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237 | (10) |
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Polynomials as Approximations |
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237 | (3) |
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Integration of Functions when Expressed as Power Series |
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240 | (2) |
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Expansion in a Series by Integrating |
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242 | (5) |
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247 | (26) |
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Definition and Properties of Complex Numbers |
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247 | (3) |
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247 | (1) |
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248 | (1) |
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248 | (1) |
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Operations with Complex Numbers |
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249 | (1) |
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Graphical Representation of Complex Numbers |
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250 | (4) |
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Gauss Complex Number Plane: Argand Diagram |
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250 | (1) |
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Polar Form of a Complex Number |
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251 | (3) |
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Exponential Form of Complex Numbers |
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254 | (7) |
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254 | (1) |
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Exponential Form of the Sine and Cosine Functions |
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255 | (1) |
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Complex Numbers as Powers |
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255 | (3) |
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Multiplication and Division in Exponential Form |
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258 | (1) |
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Raising to a Power, Exponential Form |
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259 | (1) |
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259 | (1) |
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Transformation of a Complex Number From One Form into Another |
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260 | (1) |
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Operations with Complex Numbers Expressed in Polar Form |
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261 | (12) |
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Multiplication and Division |
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261 | (2) |
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263 | (1) |
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Roots of a Complex Number |
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263 | (10) |
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273 | (48) |
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Concept and Classification of Differential Equations |
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273 | (4) |
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277 | (2) |
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General Solution of First-and Second-Order DEs with Constant Coefficients |
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279 | (12) |
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279 | (6) |
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Non-Homogeneous Linear DE |
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285 | (6) |
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291 | (2) |
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291 | (1) |
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291 | (2) |
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293 | (9) |
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293 | (1) |
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294 | (8) |
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General Linear First-Order DEs |
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302 | (4) |
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Solution by Variation of the Constant |
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302 | (2) |
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A Straightforward Method Involving the Integrating Factor |
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304 | (2) |
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Some Remarks on General First-Order DEs |
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306 | (7) |
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306 | (1) |
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307 | (1) |
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308 | (3) |
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The Integrating Factor - General Case |
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311 | (2) |
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313 | (4) |
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Higher-Order DEs Interpreted as Systems of First-Order Simultaneous DEs |
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317 | (1) |
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Some Advice on Intractable DEs |
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317 | (4) |
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321 | (16) |
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321 | (1) |
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The Laplace Transform Definition |
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321 | (1) |
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Laplace Transform of Standard Functions |
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322 | (6) |
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Solution of Linear DEs with Constant Coefficients |
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328 | (2) |
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Solution of Simultaneous DEs with Constant Coefficients |
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330 | (7) |
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Functions of Several Variables; Partial Differentiation; and Total Differentiation |
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337 | (40) |
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337 | (1) |
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Functions of Several Variables |
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338 | (6) |
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Representing the Surface by Establishing a Table of Z-Values |
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339 | (1) |
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Representing the Surface by Establishing Intersecting Curves |
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340 | (3) |
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Obtaining a Functional Expression for a Given Surface |
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343 | (1) |
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344 | (6) |
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Higher Partial Derivatives |
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348 | (2) |
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350 | (8) |
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Total Differential of Functions |
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350 | (4) |
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Application: Small Tolerances |
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354 | (2) |
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356 | (2) |
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358 | (3) |
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358 | (2) |
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360 | (1) |
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Maxima and Minima of Functions of Two or More Variables |
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361 | (6) |
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Applications: Wave Function and Wave Equation |
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367 | (10) |
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367 | (4) |
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371 | (6) |
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Multiple Integrals; Coordinate Systems |
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377 | (24) |
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377 | (2) |
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Multiple Integrals with Constant Limits |
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379 | (3) |
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Decomposition of a Multiple Integral into a Product of Integrals |
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381 | (1) |
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Multiple Integrals with Variable Limits |
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382 | (4) |
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386 | (9) |
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387 | (2) |
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389 | (2) |
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391 | (4) |
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Application: Moments of Inertia of a Solid |
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395 | (6) |
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Transformation of Coordinates; Matrices |
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401 | (28) |
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401 | (3) |
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Parallel Shift of Coordinates: Translation |
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404 | (3) |
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407 | (6) |
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407 | (3) |
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410 | (1) |
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Rotations in Three-Dimensional Space |
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411 | (2) |
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413 | (6) |
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Addition and Subtraction of Matrices |
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415 | (1) |
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Multiplication of a Matrix by a Scalar |
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416 | (1) |
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Product of a Matrix and a Vector |
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416 | (1) |
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Multiplication of Two Matrices |
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417 | (2) |
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Rotations Expressed in Matrix Form |
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419 | (2) |
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Rotation in Two-Dimensional Space |
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419 | (1) |
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Special Rotation in Three-Dimensional Space |
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420 | (1) |
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421 | (3) |
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424 | (5) |
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Sets of Linear Equations; Determinants |
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429 | (22) |
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429 | (1) |
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429 | (9) |
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Gaussian Elimination: Successive Elimination of Variables |
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429 | (2) |
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431 | (1) |
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Matrix Notation of Sets of Equations and Determination of the Inverse Matrix |
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432 | (3) |
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435 | (3) |
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438 | (13) |
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Preliminary Remarks on Determinants |
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438 | (1) |
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Definition and Properties of an n-Row Determinant |
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439 | (5) |
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Rank of a Determinant and Rank of a Matrix |
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444 | (1) |
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Applications of Determinants |
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445 | (6) |
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Eigenvalues and Eigenvectors of Real Matrices |
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451 | (10) |
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Two Case Studies: Eigenvalues of 2 x 2 Matrices |
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451 | (3) |
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General Method for Finding Eigenvalues |
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454 | (2) |
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Worked Example: Eigenvalues of a 3 x 3 Matrix |
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456 | (2) |
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Important Facts on Eigenvalues and Eigenvectors |
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458 | (3) |
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Vector Analysis: Surface Integrals, Divergence, Curl and Potential |
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461 | (30) |
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Flow of a Vector Field Through a Surface Element |
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461 | (3) |
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464 | (2) |
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Special Cases of Surface Integrals |
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466 | (4) |
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Flow of a Homogeneous Vector Field Through a Cuboid |
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466 | (2) |
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Flow of a Spherically Symmetrical Field Through a Sphere |
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468 | (2) |
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Application: The Electrical Field of a Point Charge |
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470 | (1) |
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General Case of Computing Surface Integrals |
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470 | (5) |
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Divergence of a Vector Field |
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475 | (3) |
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478 | (2) |
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480 | (4) |
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484 | (1) |
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Potential of a Vector Field |
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485 | (3) |
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Short Reference on Vector Derivatives |
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488 | (3) |
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Fourier Series; Harmonic Analysis |
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491 | (16) |
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Expansion of a Periodic Function into a Fourier Series |
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491 | (5) |
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Evaluation of the Coefficients |
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492 | (3) |
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495 | (1) |
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Examples of Fourier Series |
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496 | (5) |
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Expansion of Functions of Period 2L |
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501 | (1) |
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502 | (5) |
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507 | (12) |
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507 | (1) |
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508 | (7) |
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Random Experiment, Outcome Space and Events |
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508 | (1) |
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The Classical Definition of Probability |
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509 | (1) |
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The Statistical Definition of Probability |
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509 | (2) |
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General Properties of Probabilities |
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511 | (2) |
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Probability of Statistically Independent Events. Compound Probability |
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513 | (2) |
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Permutations and Combinations |
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515 | (4) |
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515 | (1) |
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516 | (3) |
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Probability Distributions |
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519 | (18) |
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Discrete and Continuous Probability Distributions |
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519 | (6) |
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Discrete Probability Distributions |
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519 | (3) |
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Continuous Probability Distributions |
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522 | (3) |
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Mean Values of Discrete and Continuous Variables |
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525 | (2) |
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The Normal Distribution as the Limiting Value of the Binomial Distribution |
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527 | (10) |
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Properties of the Normal Distribution |
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530 | (2) |
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Derivation of the Binomial Distribution |
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532 | (5) |
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537 | (20) |
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Purpose of the Theory of Errors |
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537 | (1) |
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538 | (4) |
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538 | (1) |
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Variance and Standard Deviation |
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539 | (1) |
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Mean Value and Variance in a Random Sample and Parent Population |
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540 | (2) |
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Mean Value and Variance of Continuous Distributions |
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542 | (2) |
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544 | (1) |
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Normal Distribution: Distribution of Random Errors |
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545 | (1) |
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546 | (2) |
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548 | (1) |
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Curve Fitting: Method of Least Squares, Regression Line |
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549 | (3) |
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Correlation and Correlation Coefficient |
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552 | (5) |
| Answers |
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557 | (24) |
| Index |
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581 | |
