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前言 CONTENTS
§5.1Infinite Series 1 §5.1.1The Concept of Infinite Series1 §5.1.2Conditions for Convergence3 §5.1.3Properties of Series5 Exercise 5.18 §5.2Tests for Convergence of Positive Series9 Exercise 5.218 §5.3Alternating Series, Absolute Convergence, and Conditional Convergence19 §5.3.1Alternating Series19 §5.3.2Absolute Convergence and Conditional Convergence21 Exercise 5.323 §5.4Tests for Improper Integrals24 §5.4.1Tests for the Improper Integrals: Infinite Limits of Integration24 §5.4.2Tests for the Improper Integrals: Infinite Integrands 26 §5.4.3The Gamma Function28 Exercise 5.430 §5.5Infinite Series of Functions31 §5.5.1General Definitions31 §5.5.2Uniform Convergence of Series32 §5.5.3Properties of Uniformly Convergent Functional Series 34 Exercise 5.536 §5.6Power Series37 §5.6.1The Radius and Interval of Convergence37 §5.6.2Properties of Power Series41 §5.6.3Expanding Functions into Power Series45 Exercise 5.655 §5.7Fourier Series56 §5.7.1The Concept of Fourier Series56 §5.7.2Fourier Sine and Cosine Series62 §5.7.3Expanding Functions with Arbitrary Period65 Exercise 5.768 Review and Exercise69 Chapter 6Vectors and Analytic Geometry in Space72 §6.1Vectors72 §6.1.1Vectors72 §6.1.2Linear Operations on Vectors73 §6.1.3Dot Products and Cross Product75 Exercise 6.179 §6.2Operations on Vectors in Cartesian Coordinates in Three Space 80 §6.2.1Cartesian Coordinates in Three Space80 §6.2.2Operations on Vectors in Cartesian Coordinates84 Exercise 6.288 §6.3Planes and Lines in Space89 §6.3.1Equations for Plane89 §6.3.2Lines92 §6.3.3Some Problems Related to Lines and Planes95 Exercise 6.3100 §6.4Curves and Surfaces in Space101 §6.4.1Sphere and Cylinder101 §6.4.2Curves in Space103 §6.4.3Surfaces of Revolution105 §6.4.4Quadric Surfaces106 Exercise 6.4109 Exercise Review110 Chapter 7Multivariable Functions and Partial Derivatives113 §7.1Functions of Several Variables113 Exercise 7.1116 §7.2Limits and Continuity116 Exercise 7.2120 §7.3Partial Derivative121 §7.3.1Partial Derivative121 §7.3.2Second Order Partial Derivatives123 Exercise 7.3126 §7.4Differentials128 Exercise 7.4132 §7.5Rules for Finding Partial Derivative133 §7.5.1The Chain Rule133 §7.5.2Implicit Differentiation137 Exercise 7.5140 §7.6Direction Derivatives, Gradient Vectors142 §7.6.1Direction Derivatives142 §7.6.2Gradient Vectors144 Exercise 7.6146 §7.7Geometric Applications of Differentiation of Functions of Several Variables147 §7.7.1Tangent Line and Normal Plan to a Curve147 §7.7.2Tangent Plane and Normal Line to a Surface149 Exercise 7.7152 §7.8Taylor Formula for Functions of Two Variables and Extreme Values 153 §7.8.1Taylor Formula for Functions of Two Variables153 §7.8.2Extreme Values155 §7.8.3Absolute Maxima and Minima on Closed Bounded Regions 160 §7.8.4Lagrange Multipliers161 Exercise 7.8164 Exercise Review166 Chapter 8Multiple Integrals172 §8.1Concept and Properties of Multiple Integrals172 §8.2Evaluation of Double Integrals174 §8.2.1Double Integrals in Rectangular Coordinates174 §8.2.2Double Integrals in Polar Coordinates178 §8.2.3Substitutions in Double Integrals182 Exercise 8.2185 §8.3Evaluation of Triple Integrals188 §8.3.1Triple Integrals in Rectangular Coordinates188 §8.3.2Triple Integrals in Cylindrical and Spherical Coordinates 192 Exercise 8.3196 §8.4Evaluation of Line Integral with Respect to Arc Length197 Exercise 8.4199 §8.5Evaluation of Surface Integrals with Respect to Area200 §8.5.1Surface Area200 §8.5.2Evaluation of Surface Integrals with Respect to Area202 Exercise 8.5204 §8.6Application for the Integrals205 Exercise 8.6208 Review and Exercise209 Chapter 9Integration in Vectors Field213 §9.1Vector Fields213 Exercise 9.1215 §9.2Line Integrals of the Second Type216 §9.2.1The Concept and Properties of the Line Integrals of the Second Type216 §9.2.2Calculation218 §9.2.3The Relation between the Two Line Integrals221 Exercise 9.2221 §9.3Green Theorem in the Plane222 §9.3.1Green Theorem223 §9.3.2Path Independence for the Plane Case228 Exercise 9.3232 §9.4The Surface Integral for Flux234 §9.4.1Orientation234 §9.4.2The Conception of the Surface Integral for Flux235 §9.4.3Calculation237 §9.4.4The Relation between the Two Surface Integrals240 Exercise 9.4241 §9.5Gauss Divergence Theorem242 Exercise 9.5246 §9.6Stoke Theorem247 §9.6.1Stoke Theorem247 §9.6.2Path Independence in Threespace251 Exercise 9.6252 Review and Exercise252 Chapter 10Complex Analysis255 §10.1Complex Numbers255 Exercise 10.1257 §10.2Complex Functions 259 §10.2.1Complex Valued Functions259 §10.2.2Limits259 §10.2.3Continuity261 Exercise 10.2263 §10.3Differential Calculus of Complex Functions264 §10.3.1Derivatives264 §10.3.2Analytic Functions268 §10.3.3Elementary Functions272 Exercise 10.3276 §10.4Complex Integration279 §10.4.1Complex Integration279 §10.4.2CauchyGoursat Theorem and Deformation Theorem 282 §10.4.3Cauchy Integral Formula and Cauchy Integral Formula for Derivatives289 Exercise 10.4292 §10.5Series Expansion of Complex Function 295 §10.5.1Sequences of Functions296 §10.5.2Taylor Series297 §10.5.3Laurent Series299 Exercise 10.5304 §10.6Singularities and Residue307 §10.6.1Singularities and Poles307 §10.6.2Cauchy Residue Theorem311 §10.6.3Evaluation of Real Integrals316 Exercise 10.6320 Exercise Review323
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