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Chapter 5 Infinite Series 5.1 Convergent Series 5.1.1 Concepts of Convergent Series 5.1.2 Properties of Convergent Series 5.2 Tests of Convergence and Divergence 5.2.1 Tests for Positive Series 5.2.2 Alternating Series 5.2.3 Absolute and Conditional Convergence 5.3 Tests for Improper Integrals 5.3.1 Improper Integrals: Infinite Limits of Integration 5.3.2 Improper Integrals.Infinite Integrands 5.3.3 The Gamma Function 5.4 Infinite Series o{ Functions 5.4.1 General Definition 5.4 .2 Uniform Convergence of Series 5.4.3 Properties of Uniformly Convergent Functional Series 5.5 Power Series 5.5 .I The Radius and Interval of Convergence 5.5.2 Properties of Power Series 5.6 Expanding Functions into Power Series 5.7 Fourier's Series 5.7.1 The Concept of Fourier's Series 5.7.2 Fourier's Sine and Cosine Series 5.7.3 Expending Functions with Arbitrary Period Chapter Review Exercise Chapter 6 Vectors and Analytic Geometry in Space 6.1 Vectors and Their Linear Operations 6.1.1 The Concept of Vector 6.1.2 Linear Operations on Vectors 6.1.3 Dot Product and Gross Product 6.2 Operations on Vectors in Cartesian Coordinates in Three Space 6.2.1 Cartesian Coordinates in Three Space 6.2.2 Operations on Vectors in Cartesian Coordinates 6.3 Planes and Lines in Space 6.3.1 Equations for Plane 6.3.2 Lines 6.3.3 Some Problems Related to Lines and Planes 6.4 Curves and Surfaces in Space 6.4.1 Sphere and Cylinder 6.4.2 Curves in Space 6,4.3 Cone 6.4.4 Surfaces of Revolution 6.4.5 Quadric Surfaces 6.5 Vector-Valued Functions 6.5.1 Limit of a Vector-Valued Function 6.5.2 Derivative of a Vector-Valued Function 6.5.3 Integral of a Vector-Valued Function 6.5.4 Curvilinear Motion 6.5.5 Curvature Chapter Review Exercise Chapter 7 Multivariate Functions and Partial Derivatives 7.1 Functions of Several Variables 7.2 Limits and Continuity 7.3 Partial Derivative 7.3.1 Definitions of Partial Derivative 7.3.2 Geometric Interpretation of Partial Derivative 7.3.3 Second Order Partial Derivatives 7.4 Differentials 7.5 Rules for Finding Partial Derivative 7.5.1 The Chain Rule 7.5.2 Implicit Differentiation 7.6 Directional Derivatives and Gradient Vectors 7.6.1 Directional Derivatives 7.6.2 Gradient Vectors 7.7 Geometric Applications of Differentiation of Functions of Several Variables 7.7.1 Tangent Line and Normal Plane to a Curve 7.7.2 Tangent Plane and Normal Line to a Surface 7.8 Taylor's Formula for Functions of Two Variables and Extreme Values 7.8.1 Taylor's Formula for Functions of Two Variables 7.8.2 Extreme Values 7.8.3 Absolute Maxima and Minima on Closed Bounded Regions 7.8.4 Lagrange Multipliers Chapter Review Exercise Chapter 8 Multiple Integrals 8.1 Concept and Properties of Integrals 8.1.1 Concept of the Multiple Integrals 8.1.2 Properties of the Multiple Integrals 8.2 Evaluation of Double Integrals 8.2.1 Double Integrals in Rectangular Coordinates 8.2.2 Double Integrals in Polar Coordinates 8.2.3 Substitutions in Double Integrals 8.3 Evaluation of Triple Integrals 8.3.1 Triple Integrals in Rectangular Coordinates 8.3.2 Triple Integrals in Cylindrical and Spherical Coordinates 8.4 Line Integrals with Respect to Arc Length 8.5 Surface Integrals with Respect to Area 8.5.1 Surface Area 8.5.2 Evaluation of Surface Integrals with Respect to Area 8.6 Application for the Integrals Chapter Review Exercise Chapter 9 Integration in Vector Fields 9.1 Vector Fields 9.2 Line Integrals of the Second Type 9.2.1 The Concept and Properties of the Line Integrals of the Second Type 9.2.2 Calculation 9.2.3 The Relation between the Two Line Integrals 9.3 Green's Theorem in the Plane 9.3.1 Green's Theorem 9.3.2 Path Independence for the Plane Case 9.4 The Surface Integral for Flux 9.4.1 Orientation 9.4.2 The Conception of the Surface Integral for Flux 9.4.3 Calculation 9.4.4 The Relation between the Two Surface Integrals 9.5 Gauss's Divergence Theorem 9.6 Stokes's Theorem Chapter Review Exercise Solutions to Selected Problem