Chapter 5 Infinite Series

5.1 Infinite Series

5.1.1 The Concept of Infinite Series

5.1.2 Conditions for Convergence

5.1.3 Properties of Series

Exercise 5.1

5.2 Tests for Convergence of Positive Series

Exercise 5.2

5.3 Alternating Series,Absolute Convergence,and Conditional Convergence

5.3.1 Alternating Series

5.3.2 Absolute Convergence and Conditional Convergence

Exercise 5.3

5.4 Tests for Improper Integrals

5.4.1 Tests for the Improper Integrals:Infinite Limits of Integration

5.4.2 Tests for the Improper Integrals:Infinite Integrands

5.4.3 The Gamma Function

Exercise 5.4

5.5 Infinite Series of Functions

5.5.1 General Definitions

5.5.2 Uniform Convergence of Series

5.5.3 Properties of Uniformly Convergent Functional Series

Exercise 5.5

5.6 Power Series

5.6.1 The Radius and Interval of Convergence

5.6.2 Properties of Power Series

5.6.3 Expanding Functions into Power Series

Exercise 5.6

5.7 Fourier Series

5.7.1 The Concept of Fourier Series

5.7.2 Fourier Sine and Cosine Series

5.7.3 Expanding Functions with Arbitrary Period

Exercise 5.7

Review and Exercise

Chapter 6 Vectors and Analytic Geometry in Space

6.1 Vectors

6.1.1 Vectors

6.1.2 Linear Operations on Vectors

6.1.3 Dot Products and Cross Product

Exercise 6.1

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

Exercise 6.2

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

Exercise 6.3

6.4 Curves and Surfaces in Space

6.4.1 Sphere and Cylinder

6.4.2 Curves in Space

6.4.3 Surfaces of Revolution

6.4.4 Quadric Surfaces

Exercise 6.4

Exercise Review

Chapter 7 Multivariable Functions and Partial Derivatives

7.1 Functions of Several Variables

Exercise 7.1

7.2 Limits and Continuity

Exercise 7.2

7.3 Partial Derivative

7.3.1 Partial Derivative

7.3.2 Second Order Partial Derivatives

Exercise 7.3

7.4 Differentials

Exercise 7.4

7.5 Rules for Finding Partial Derivative

7.5.1 The Chain Rule

7.5.2 Implicit Differentiation

Exercise 7.5

7.6 Direction Derivatives,Gradient Vectors

7.6.1 Direction Derivatives

7.6.2 Gradient Vectors

Exercise 7.6

7.7 Geometric Applications of Differentiation of Functions of Several Variables

7.7.1 Tangent Line and Normal Plan to a Curve

7.7.2 Tangent Plane and Normal Line to a Surface

Exercise 7.7

7.8 Taylor Formula for Functions of Two Variables and Extreme Values

7.8.1 Taylor 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

Exercise 7.8

Exercise Review

Chapter 8 Multiple Integrals

8.1 Concept and Properties of 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

Exercise 8.2

8.3 Evaluation of Triple Integrals

8.3.1 Triple Integrals in Rectangular Coordinates

8.3.2 Triple Integrals in Cylindrical and Spherical Coordinates

Exercise 8.3

8.4 Evaluation of Line Integral with Respect to Arc Length

Exercise 8.4

8.5 Evaluation of Surface Integrals with Respect to Area

8.5.1 Surface Area

8.5.2 Evaluation of Surface Integrals with Respect to Area

Exercise 8.5

8.6 Application for the Integrals

Exercise 8.6

Review and Exercise

Chapter 9 Integration in Vectors Field

9.1 Vector Fields

Exercise 9.1

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

Exercise 9.2

9.3 Green Theorem in the Plane

9.3.1 Green Theorem

9.3.2 Path Independence for the Plane Case

Exercise 9.3

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

Exercise 9.4

9.5 Gauss Divergence Theorem

Exercise 9.5

9.6 Stoke Theorem

9.6.1 Stoke Theorem

9.6.2 Path Independence in Three-space

Exercise 9.6

Review and Exercise

Chapter 10 Complex Analysis

10.1 Complex Numbers

Exercise 10.1

10.2 Complex Functions

10.2.1 Complex Valued Functions

10.2.2 Limits

10.2.3 Continuity

Exercise 10.2

10.3 Differential Calculus of Complex Functions

10.3.1 Derivatives

10.3.2 Analytic Functions

10.3.3 Elementary Functions

Exercise 10.3

10.4 Complex Integration

10.4.1 Complex Integration

10.4.2 Cauchy-Goursat Theorem and Deformation Theorem

10.4.3 Cauchy Integral Formula and Cauchy Integral Formula for Derivatives

Exercise 10.4

10.5 Series Expansion of Complex Function

10.5.1 Sequences of Functions

10.5.2 Taylor Series

10.5.3 Laurent Series

Exercise 10.5

10.6 Singularities and Residue

10.6.1 Singularities and Poles

10.6.2 Cauchy Residue Theorem

10.6.3 Evaluation of Real Integrals

Exercise 10.6

Exercise Review