CHAPTER 5 Vecto and the Geometry of Space
5.1 Vecto
5.l.1 Concepts of Vecto
5.l.2 Linear Operatio Involving Vecto
5.1.3 Coordinate Systems in Three-Dimeional Space
5.1.4 Representing Vecto Using Coordinates
5.1.5 Lengths, Direction Angles and Projectio of Vecto
5.2 Dot Product, Cross Product and Scalar Triple Product
5.2.l The Dot Product
5.2.2 The Cross Product
5.2.3 Scalar Triple Product
5.3 Equatio of Planes and Lines
5.3.1 Planes
5.3.2 Lines
5.4 Surfaces In Space
5.4.1 Surfaces and Equatio
5.4.2 Cylinder
5.4.3 Surface of Revolution
5.4.4 Quadric Surfaces
5.5 Curves in Space
5.5.1 General Equatio of Curves in the Space
5.5.2 Parametric Equatio of Curves in the Space
5.5.3 Parametric Equatio of Surfaces in the Space
5.5.4 Projectio of Curves in the Space
5.6 Exercises
5.6.1 Vecto
5.6.2 Planes and Lines in Space
5.6.3 Surfaces and Curves in Space
5.6.4 Questio to Guide Your Revision
CHAPTER 6 Functio of Several Variables
6.1 Functio 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 Tota Differential
6.3.1 Definition
6.3.2 The Total Differential Approximation
6.4 The Chain Rule
6.5 Implicit Differentiation
6.5.1 Functio Defined by a Single Equation
6.5.2 Functio Defined Implicitly by System of Equatio
6.6 Applicatio of the Differential Calculus
6.6.I Tangent Lines and Normal Planes
6.6.2 Tangent Planes and Normal Lines for Surfaces
6.7 Directional Derivatives and Gradient Vecto
6.8 Maximum and Minimum
6.8.l Extrema of Functio of Several Variables
6.8.2 Lagrange Multiplie
6.9 Additional Materials
6.9.1 Taylor''s Theorem for Functio of Two Variables
6.9.2 Clairaut
6.9.3 Cobb-Douglas Production Function
6.10 Exercises
6.10.1 Functio of Several Variables
6.10.2 Applicatio of Partial, Derivatives
6.10.3 Questio 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 Integrals
7.6.2 Triple Integrals
7.6.3 Applicatio of Multiple Integrals
7.6.4 Questio to Guide Your Revision
CHAPTER 8 Line and SUrface Integrals
8.1 Line Integrals
8.l.1 Introduction
8.1.2 Definition of the Line Integral with ReSpect to Arc
Length
8.1.3 Evaluating Line Integrals, ffx, yas, in R2
8.1.4 Evaluating Line Integrals, ffx, y, zds, 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 fcF·dr
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 Coervative 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,ffsF·NdS
8.6.2 Evaluating ffF·NdS foraSurfacez=zx, y
8.7 The Divergence Theorem
8.7.1 Introduction
8.7.2 Physical interpretation of the Divergence V· Fx, y, z
8.8 Stoke''s Theorem
8.9 Additional Materials
8.9.I 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 Questio to Guide Your Revision
CHAPTER 9 Infinite Sequences, Series and Approximatio
9.1 Infinite Sequences
9.2 Infinite Series
9.2.1 Definition of Infinite Serie
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 Applicatio of Power Series
9.5 Fourier Series
9.5.1 Fourier Series Expaion with Period 2π
9.5.2 Fourier Cosine and Sine Series with Period 2π
9.5.3 The Fourier Series Expaion 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 Cotant Terms
9.7.2 Power Series
9.7.3 Fourier Series
9.7.4 Questio to Guide Your Revision
CHAPTER 10 Introduction to Ordinary Differential Equation
10.1 Differential Equatio and Mathematical Models
10.2 Methods for Solving Ordinary Differential Equatio
10.2.1 Separable Equatio
10.2.2 Substitution Methods
10.2.3 Exact Differential Equatio
10.2.4 Linear Fit-Order Differential Equatio and Integrating
Facto .-
10.2.5 Reducible Second-Order Equatio
10.2.6 Linear Second-Order Differential Equatio
10.3 Other Ways of Solving Differential Equatio
I0.3.1 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 Equatio
10.5.2 Fit Order Differential Equation
10.5.3 Second Order Differential Equation
10.5.4 Questio to Guide Your Revision
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