Chapter 8 Vector Algebra and Analytic Geometry of Space
　　 8.1 Vectors and their linear operations
　　 8.1.1 The concept of vector
　　 8.1.2 Vector linear operations
　　 8.1.3 Three-dimensional rectangular coordinate system
　　 8.1.4 Component representation of vector linear operations
　　 8.1.5 Length，direction angles and projection of a vector
　　 Exercise 8-1
　　 8.2 Multiplicative operations on vectors
　　 8.2.1 The scalar product（dot product，inner product）of two vectors
　　 8.2.2 The vector product（cross product，outer product）of two vectors
　　 *8.2.3 The mixed product of three vectors
　　 Exercise 8-2
　　 8.3 Surfaces and their equations
　　 8.3.1 Definition of surface equations
　　 8.3.2 Surfaces of revolution
　　 8.3.3 Cylinders
　　 8.3.4 Quadric surfaces
　　 Exercise 8-3
　　 8.4 Space curves and their equations
　　 8.4.1 General form of equations of space curves
　　 8.4.2 Parametric equations of space curves
　　 *8.4.3 Parametric equations of a surface
　　 8.4.4 Projections of space curves on coordinate planes
　　 Exercise 8-4
　　 8.5 Plane and its equation
　　 8.5.1 Point-normal form of the equation of a plane
　　 8.5.2 General form of the equation of a plane
　　 8.5.3 The included angle between two planes
　　 Exercise 8-5
　　 8.6 Straight line in space and its equation
　　 8.6.1 General form of the equations of a straight line
　　 8.6.2 Parametric equations and symmetric form equations of a straight line
　　 8.6.3 The included angel between two lines
　　 8.6.4 The included angle between a line and a plane
　　 8.6.5 Some examples
　　 Exercise 8-6
　　Exercise 8
　　Chapter 9 The multivariable differential calculus and its applications44
　　 9.1 Basic concepts of multivariable functions
　　 9.1.1 Planar sets n-dimensional space
　　 9.1.2 The concept of a multivariable function
　　 9.1.3 Limits of multivariable functions
　　 9.1.4 Continuity of multivariable functions
　　 Exercise 9-1
　　 9.2 Partial derivatives
　　 9.2.1 Definition and computation of partial derivatives
　　 9.2.2 Higher-order partial derivatives
　　 Exercise 9-2
　　 9.3 Total differentials
　　 9.3.1 Definition of total differential
　　 9.3.2 Applications of the total differential to approximate computation
　　 Exercise 9-3
　　 9.4 Differentiation of multivariable composite functions
　　 9.4.1 Composition of functions of one variable and multivariable functions
　　 9.4.2 Composition of multivariable functions and multivariable functions
　　 9.4.3 Other case
　　 Exercise 9-4
　　 9.5 Differentiation of implicit functions
　　 9.5.1 Case of one equation
　　 9.5.2 Case of system of equations
　　 Exercise 9-5
　　 9.6 Applications of differential calculus of multivariable functions in geometry
　　 9.6.1 Derivatives and differentials of vector-valued functions of one variable
　　 9.6.2 Tangent line and normal plane to a space curve
　　 9.6.3 Tangent plane and normal line of surfaces
　　 Exercise 9-6
　　 9.7 Directional derivatives and gradient
　　 9.7.1 Directional derivatives
　　 9.7.2 Gradient
　　 Exercise 9-7
　　 9.8 Extreme value problems for multivariable functions
　　 9.8.1 Unrestricted extreme values and global maxima and minima
　　 9.8.2 Extreme values with constraints the method of Lagrange multipliers
　　 Exercise 9-8
　　 9.9 Taylor formula for functions of two variables
　　 9.9.1 Taylor formula for functions of two variables
　　 9.9.2 Proof of the sufficient condition for extreme values of function of two variables
　　 Exercise 9-9
　　Exercise 9
　　Chapter 10 Multiple Integrals
　　 10.1 The concept and properties of double integrals
　　 10.1.1 The concept of double integrals
　　 10.1.2 Properties of double Integrals
　　 Exercise 10-1
　　 10.2 Computation of double integrals
　　 10.2.1 Computation of double integrals in rectangular coordinates
　　 10.2.2 Computation of double integrals in polar coordinates
　　 *10.2.3 Integration by substitution for double integrals
　　 Exercise 10-2
　　 10.3 Triple integrals
　　 10.3.1 Concept of triple integrals
　　 10.3.2 Computation of triple integrals
　　 Exercise 10-3
　　 10.4 Application of multiple integrals
　　 10.4.1 Area of a surface
　　 10.4.2 Center of mass
　　 10.4.3 Moment of inertia
　　 10.4.4 Gravitational force
　　 Exercise 10-4
　　 10.5 Integral with parameter
　　 Exercise 10-5
　　Exercise 10
　　Chapter 1 1Line and Surface Integrals
　　 11.1 Line integrals with respect to arc lengths
　　 11.1.1 The concept and properties of the line integral with respect to arc lengths
　　 11.1.2 Computation of line integral with respect to arc lengths
　　 Exercise 11-1
　　 11.2 Line integrals with respect to coordinates
　　 11.2.1 The concept and properties of the line integrals with respect to coordinates
　　 11.2.2 Computation of line integrals with respect to coordinates
　　 11.2.3 The relationship between the two types of line integral
　　 Exercise 11-2
　　 11.3 Green’s formula and the application to fields
　　 11.3.1 Green’s formula
　　 11.3.2 The conditions for a planar line integral to have independence of path
　　 11.3.3 Quadrature problem of the total differential
　　 Exercise 11-3
　　 11.4 Surface integrals with respect to acreage
　　 11.4.1 The concept and properties of the surface integral with respect to acreage
　　 11.4.2 Computation of surface integrals with respect to acreage
　　 Exercise 11-4
　　 11.5 Surface integrals with respect to coordinates
　　 11.5.1 The concept and properties of the surface integrals with respect to coordinates
　　 11.5.2 Computation of surface integrals with respect to coordinates
　　 11.5.3 The relationship between the two types of surface integral
　　 Exercise 11-5
　　 11.6 Gauss’ formula
　　 11.6.1 Gauss’ formula
　　 *11.6.2 Flux and divergence
　　 Exercise 11-6
　　 11.7 Stokes formula
　　 11.7.1 Stokes formula
　　 11.7.2 Circulation and rotation
　　 Exercise 11-7
　　Exercise 11
　　Chapter 12 Infinite Series
　　 12.1 Concepts and properties of series with constant terms
　　 12.1.1 Concepts of series with constant terms
　　 12.1.2 Properties of convergence with series
　　 *12.1.3 Cauchy’s convergence principle
　　 Exercise 12-1
　　 12.2 Convergence tests for series with constant terms
　　 12.2.1 Convergence tests for series of positive terms
　　 12.2.2 Alternating series and Leibniz’s test
　　 12.2.3 Absolute and conditional convergence
　　 Exercise 12-2
　　 12.3 Power series
　　 12.3.1 Concepts of series of functions
　　 12.3.2 Power series and convergence of power series
　　 12.3.3 Operations on power series
　　 Exercise 12-3
　　 12.4 Expansion of functions in power series
　　 Exercise 12-4
　　 12.5 Application of expansion of functions in power series
　　 12.5.1 Approximations by power series
　　 12.5.2 Power series solutions of differential equation
　　 12.5.3 Euler formula
　　 Exercise 12-5
　　 12.6 Fourier series
　　 12.6.1 Trigonometric series and orthogonality of the system of trigonometric functions
　　 12.6.2 Expand a function into a Fourier series
　　 12.6.3 Expand a function into the sine series and cosine series
　　 Exercise 12-6
　　 12.7 The Fourier series of a function of period 21
　　 Exercise 12-7
　　Exercise 12
　　Reference