| The main subject of this book is calculusBesides，it also includes differential equation，analytic geometry in space，vector algebra and infinite seriesThis book is divided into two volumesThe first volume contains calculus of functions of a single variable and differential equationThe second volume contains vector algebra and analytic geometry in space，multivariable calculus and infinite series. |
| 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 |
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