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Preface
Translator's note
Introduction
1 Real and complex numbers
1.1 Rational numbers
1.2 The existence of irrational numbers
1.3 A description of real numbers
1.4 Limit
1.5 The Bolzano—Weierstrass theorem
1.6 Definitions for complex numbers and vectors
1.7 Polar coordinates and multiplication
1.8 De Moivre's theorem
1.9 Completeness of the complex numbers
1.10 Introduction to quaternions Supplement
1.11 Binary arithmetics
1.12 Periodic decimals
1.13 Rational approximations to real numbers
1.14 Errorterms
1.15 Solutions to cubic and quartic equations
2 Vector algebra
2.1 Space coordinates and vectors
2.2 Addition of vectors
2.3 The decomposition of a vector
2.4 Inner product （scalar product）
2.5 Vector product （outer product）
2.6 Multiple products
2.7 Change of coordinates
2.8 Planes
2.9 Equation for a line in space Supplement
2.10 Main formulae in spherical trigonometry
2.11 Dualityprinciple
2.12 Right—angled and right—sided triangles
2.13 Forces， systems and equivalent systems
2.14 Combination of parallel forces
2.15 Moments
2.16 Couples
2.17 Standard form for a system
2.18 Equilibrium and its applications
3 Functions and graphs
3.1 Variables
3.2 Functions
3.3 Implicit functions
3.4 Functions represented by graphs and tables
3.5 Several elementary functions
3.6 Functions with simple special properties
3.7 Periodic functions
3.8 Representations for a complex function
3.9 Line of regression
3.10 Lagrange's interpolation formula
3.11 Other interpolation formulae
3.12 Experimentalformulae
3.13 Family ofcurves
4 Limits
4.1 Limits of sequences
4.2 Sequences without limits
4.3 Series
4.4 Conditionally convergent series
4.5 The method of Zu Chongzhi in calculating π
4.6 Archimedes' method for the area of a parabolic region
4.7 Calculating pressure on a boundary
4.8 The number e
4.9 Taking limit in the continuum
4.10 On severalimportant limits
4.11 Some examples
4.12 Orders of infinity
4.13 The symbols～O and o
4.14 Continuous functions
4.15 Types of discontinuities
4.16 Some fundamental properties of continuous functions
4.17 The Heine—Borel theorem
5 The differential calculus
5.1 The notion of the derivative
5.2 Geometric interpretation of the derivative
5.3 Derivatives of sums and products
5.4 Derivatives of elementary functions
5.5 Derivatives of composite functions
5.6 The hyperbolic functions
5.7 Formulae for differentiation
5.8 Examples
5.9 Differentials
5.10 Errorestimates
5.11 Higherderivatives
5.12 Leibniz's formula
5.13 Higherdifferentials
5.14 Differences in functions
6 Applications of the derivative
6.1 Ups and downs along a curve
6.2 Maxima and minima
6.3 Fermat's theorem
6.4 Mean—value formula
6.5 Convexity and points ofinflection
6.6 Asymptotes
6.7 Essential points in curve sketching
6.8 Sketching parametric curves
6.9 Tangents and normals
6.10 Integrationformulae
6.11 Implicit differentiation
6.12 The indeterminate form 0／0
6.13 Theindeterminate form ∞／∞
6.14 Other indeterminate forms
7 Taylor expansions
7.1 Taylor's formula for a polynomial
7.2 Taylor expansions for functions
7.3 Taylor series and remainder terms
7.4 The expansion for ex
7.5 Expansions for sin x and cos x
7.6 The binomial expansion
7.7 The expansion for log（1+x）
7.8 The expansion for arctan x
7.9 Power series and radius of convergence
7.10 Arithmetic operations on power series
7.11 Differentiation and integration of power series
7.12 Uniqueness theorem andinverse functions
7.13 Kummer's test and Gauss' test
7.14 Hypergeometric series
7.15 Power series solutions of differential equations
8 Approximate solutions to equations
8.1 Introduction
8.2 Graphical methods
8.3 Method of successive substitutions
8.4 Interpolation method
8.5 Newton's method
8.6 A combination of methods
8.7 Digital refinement method
8.8 Lobachevskiy method Supplement
8.9 Theorems on real roots
8.10 Sturm's theorem
9 Indefinite integrals
9.1 Change of variables
9.2 Integration by parts
9.3 Partial fractions
9.4 Integration of rational functions
9.5 Ostrogradskiy method
9.6 Integration of certain functions with roots
9.7 Integration of f R（x，√ax2+bx+c）dx
9.8 Abelian integrals
9.9 Integrals not representable by 'known' functions
9.10 Differentialequations， variables separable
9.11 Homogeneous differential equations
9.12 Integrating factor method
9.13 First orderlinear equations
9.14 Second orderlinear equations
9.15 Linear equations with constant coefficients
10 Definite integrals
10.1 Area determination
10.2 The notion of a definite integral
10.3 Properties of integrable functions
10.4 Fundamental properties of definite integrals
10.5 Mean—value theorem and the fundamental theorem of calculus
10.6 The second mean—value theorem
10.7 Examples
10.8 Integration by substitution
10.9 Integration by parts
10.10 Improperintegrals
10.11 Applications of the definite integral
10.12 Integration by special techniques
10.13 Applications of area consideration
10.14 Euler's summation formula
10.15 Trapezium， rectangle and Simpson's rules
……
11 Applications of integration
12 Functions of several variables
13 Sequences， series and integrals of functions
14 Properties of differentials of curves
15 Multiple integrals
16 Line integrals and surface integrals
17 Potential fields and vector fields
18 Properties of differentials of surfaces
19 Fourier series
20 Systems of ordinary differential equations 

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