CHAPTER 5 Vectors and the Geometry of Space

5.1 Vectors

5.1.1 Concepts of Vectors

5.1.2 Linear Operations Involving Vectors

5.1.3 Coordinate Systems in Three-Dimensional Space

5.1.4 Representing Vectors Using Coordinates

5.1.5 Lengths,Direction Angles and Projections of Vectors

5.2 Dot Product,Cross Product and Scalar Triple Product

5.2.1 The Dot Product

5.2.2 The Cross Product

5.2.3 Scalar Triple Product

5.3 Equations of Planes and Lines

5.3.1 Planes

5.3.2 Lines

5.4 Surfaces In Space

5.4.1 Surfaces and Equations

5.4.2 Cylinder

5.4.3 Surface of Revolution

5.4.4 Quadric Surfaces

5.5 Curves in Space

5.5.1 General Equations of Curves in the Space

5.5.2 Parametric Equations of Curves in the Space

5.5.3 Parametric Equations of Surfaces in the Space

5.5.4 Projections of Curves in the Space

5.6 Exercises

5.6.1 Vectors

5.6.2 Planes and Lines in Space

5.6.3 Surfaces and Curves in Space

5.6.4 Questions to Guide Your Revision

CHAPTER 6 Functions of Several Variables

6.1 Functions of Several Variables

6.1.1 Definition

6.1.2 Limits

6.1.3 Continuity

6.2 Partial Derivatives

6.2.1 Definition

6.2.2 Partial Derivative of Higher Order

6.3 Total Differential

6.3.1 Definition

6.3.2 The Total Differential Approxination

6.4 The Chain Rule

6.5 Implicit Differentiation

6.5.1 Functions Defined by a Single Equation

6.5.2 Functions Defined Implicitly by System of Equations

6.6 Applications of the Differential Calculus

6.6.1 Tangent Lines and Normal Planes

6.6.2 Tangent Planes and Normal Lines for Surfaces

6.7 Directional Derivatives and Gradient Vectors

6.8 Maximum and Minimum

6.8.1 Extrema of Functions of Several Variables

6.8.2 Lagrange Multipliers

6.9 Additional Materials

6.9.1 Taylor's Theorem for Functions of Two Variables

6.9.2 Clairaut

6.9.3 Cobb-Douglas Production Function

6.10 Exercises

6.10.1 Funetions of Several Variables

6.10.2 Applications of Partial Derivatives

6.10.3 Questions to Guide Your Revision

CHAPTER 7 Multiple Integrals

7.1 Definition and Properties

7.2 Iterated Integrals

7.2.1 Iterated Integrals in Rectangular Coordinates

7.2.2 Change of Variables Formula for Double Integrals

7.3 Triple Integrals

7.3.1 Triple Integrals in Rectangular Coordinates

7.3.2 Change of Variables in Triple Integrals

7.4 The Area of a Surface

7.5 Additional Materials

7.6 Exercises

7.6.1 Double Integralsv

7.6.2 Triple Integrals

7.6.3 Applications of Multiple Integrals

7.6.4 Questions to Guide Your Revision

CHAPTER 8 Line and Surface Integrals

8.1 Line Integrals

8.1.1 Introduction

8 1.2 Definition of the Line Integral with Respect to Are Length

8.1.3 Evaluating Line Integrals,∫cf(x,y)ds,in R2

8.1.4 Evaluating Line Integrals,∫cf(x,y,z)ds,in R3

8 2 Vector Fields,Work,and Flows

8.2.1 Introduction

8.2.2 The Line Integral of a Vector Field Along a Curve C

8.2.3 Different Forms of the Line Integral Including ∫c→Fd→r

8.2.4 Examples of Line Integrals

8.3 Green's Theorem in R2

8.3.1 The Circulation-Curl Form of Green's Theorem

8.3.2 The Divergence-Flux Form of Green's Theorem

8.3.3 Generalized Green's Theorem

8.4 Path Independent Line Integrals and Conservative Fields

8.4.1 Introduction

8.4.2 Fundamental Results on Path Independent Line Integrals

8.5 Surface Integrals

8.5.1 Definition of Integration With Respect to Surface Area

8.5.2 Evaluation of Surface Integrals

8.6 Surface Integrals of Vector Fields

8.6.1 Definition and Properties of Flux,∫∫s→F→NdS

8.6.2 Evaluating ∫∫s→F·→NdS for a Surface z=z(x,y)

8.7 The Divergence Theorem

8.7.1 Introduction

8.7.2 Physical interpretation of the Divergence ？·→F(x,y,z)

8.8 Stoke's Theorem

8.9 Additional Materials

8.9.1 Green

8.9.2 Gauss

8.9.3 Stokes

8.10 Exercises

8 10.1 Line Integrals

8.10.2 Surface Integrals

8.10.3 Questions to Guide Your Revision

CHAPTER 9 Infinite Sequences,Series and Approximations

9.1 Infinite Sequences

9.2 Infinite Series

9.2.1 Definition of Infinite Series

9.2.2 Properties of Convergent Series

9.3 Tests for Convergence

9.3.1 Series with Nonnegative Terms

9.3.2 Series with Negative and Positive Terms

9.4 Power Series and Taylor Series

9.4.1 Power Series

9.4.2 Working with Power Series

9.4.3 Taylor Series

9.4.4 Applications of Power Series

9.5 Fourier Series

9.5.1 Fourier Series Expansion with Period 2π

9.5.2 Fourier Cosine and Sine Series with Period 2π

9.5.3 The Fourier Series Expansion with Period 2l

9.5.4 Fourier Series with Complex Terms

9.6 Additional Materials

9.6.1 Fourier

9.6.2 Maclaurin

9.6.3 Taylor

9.7 Exercises

9.7.1 Series with Constant Terms

9.7.2 Power Series

9.7.3 Fourier Series

9.7.4 Questions to Guide Your Revision

CHAPTER 10 Introduction to Ordinary Differential Equation

10.1 Differential Equations and Mathematical Models

10.2 Methods for Solving Ordinary Differential Equations

10.2.1 Separable Equations

10.2.2 Substitution Methods

10.2.3 Exact Differential Equations

10.2.4 Linear First-Order Differential Equations and Integrating Factors

10.2.5 Reducible Second-Order Equations

10.2.6 Linear Second-Order Differential Equations

10.3 Other Ways of Solving Differential Equations

10.3.1 Power Series Method

10.3.2 Direction Fields

10.3.3 Numerical Approximation:Euler's Method

10.4 Additional Materials

10.4.1 Euler

10.4.2 Bernoulli

10.4.3 The Bernoulli Family

10.4.4 Development of Calculus

10.5 Exercises

10.5.1 Introduction to Differential Equations

10.5.2 First Order Differential Equation

10.5.3 Second Order Differential Equation

10.5.4 Questions to Guide Your Revision

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