英文共同题名：Advanced mathematics

chapter 0 preliminary knowledge
0.1 polar coordinate system
　0.1.1 plotting points with polar coordinates
　 0.1.2 converting between polar and cartesian
coordinates
0.2 complex numbers
　 0.2.1 the definition of the complex number
　0.2.2 the complex plane
　 0.2.3 absolute value,conjugation and distance
　0.2.4 polar form of complex numbers
chapter 1 theoretical basis of calculus
1.1 sets and functions
　 1.1.1 sets and their operations
　 1.1.2 mappings and functions
　1.1.3 the primary properties of functions
　1.1.4 composition of functions
　 1.1.5 elementary functions and hyperbolic functions
　1.1.6 modeling our real world
　 exercises 1.1
1.2 limits of sequences of numbers
　1.2.1 the sequence
　 1.2.2 convergence of a sequence
　1.2.3 calculating limits of sequences
　 exercises 1.2
1.3 limits of functions
　1.3.1 speed and rates of change
　 1.3.2 the concept of limit of a function
　1.3.3 properties and operation rules of functional
limits
　 1.3.4 two important limits
　exercises 1.3
1.4 infinitesimal and infinite quantities
　 1.4.1 infinitesimal quantities and their order
　1.4.2 infinite quantities
　 exercises 1.4
1.5 continuous functions
　1.5.1 continuous function and discontinuous points
　 1.5.2 operations on continuous functions and the continuity
of elementary functions
　 1.5.3 properties of continuous functions on a closed
interval
　exercises 1.5
chapter 2 derivative and differential
2.1 concept of derivatives
　 2.1.1 introductory examples
　2.1.2 definition of derivatives
　 2.1.3 geometric interpretation of derivative
　2.1.4 relationship between derivability and
continuity
　 exercises 2.1
2.2 rules of finding derivatives
　2.2.1 derivation rules of rational operations
　 2.2.2 derivative of inverse functions
　2.2.3 derivation rules of composite functions
　 2.2.4 derivation formulas of fundamental elementary
functions
　exercises 2.2
2.3 higher-order derivatives
　 exercises 2.3
2.4 derivation of implicit functions and parametric
equations,related rates
　2.4.1 derivation of implicit functions
　 2.4.2 derivation of parametric equations
　2.4.3 related rates
　 exercises 2.4
2.5 differential of the function
　2.5.1 concept of the differential
　 2.5.2 geometric meaning of the differential
　2.5.3 differential rules of elementary functions
　 exercises 2.5
2.6 differential in linear approximate computation
　exercises 2.6
chapter 3 the mean value theorem and applications of
derivatives
3.1 the mean value theorem
　 3.1.1 rolle's theorem
　 3.1.2 lagrange's theorem
　3.1.3 cauchy s theorem
　 exercises 3.1
3.2 l'hospital's rule
　 exercises 3.2
3.3 taylor's theorem
　3.3.1 taylor's theorem
　 3.3.2 applications of taylor's theorem
　exercises 3.3
3.4 monotonicity and convexity of functions
　 3.4.1 monotonicity of functions
　3.4.2 convexity of functions,inflections
　 exercises 3.4
3.5 local extreme values,global maxima and minima
　3.5.1 local extreme values
　 3.5.2 global maxima and minima
　 exercises 3.5
3.6 graphing functions using calculus
　 exercises 3.6
chapter 4 indefinite integrals
4.1 concepts and properties of indefinite integrals
　 4.1.1 antiderivatives and indefinite integrals
　 4.1.2 properties of indefinite integrals
　 exercises 4.1
4.2 integration by substitution
　4.2.1 integration by the first substitution
　 4.2.2 integration by the second substitution
　exercises 4.2
4.3 integration by parts
　 exercises 4.3
4.4 integration of rational fractions
　 4.4.1 integration of rational fractions
　 4.4.2 antiderivatives not expressed by elementary
functions
　exercises 4.4
chapter 5 definite integrals
5.1 concepts and properties of definite integrals
　 5.1.1 instances of definite integral problems
　 5.1.2 the definition of definite integral
　5.1.3 properties of definite integrals
　 exercises 5.1
5.2 the fundamental theorems of calculus
　exercises 5.2
5.3 integration by substitution and by parts in definite
integrals
　 5.3.1 substitution in definite integrals
　 5.3.2 integration by parts in definite integrals
　exercises 5.3
5.4 improper integral
　 5.4.1 integration on an infinite interval
　5.4.2 improper integrals with infinite
discontinuities
　 exercises 5.4
5.5 applications of definite integrals
　5.5.1 method of setting up elements of integration
　 5.5.2 the area of a plane region
　 5.5.3 the arc length of a curve
　 5.5.4 the volume of a solid
　 5.5.5 applications of definite integral in physics
　 exercises 5.5
chapter 6 infinite series
6.1 concepts and properties of series with constant
terms
　 6.1.1 examples of the sum of an infinite sequence)
　6.1.2 concepts of series with constant terms
　 6.1.3 properties of series with constant terms
　exercises 6.1
6.2 convergence tests for series with constant terms
　 6.2.1 convergence tests of series with positive terms
　6.2.2 convergence tests for alternating series
　 6.2.3 absolute and conditional convergence
　exercises 6.2
6.3 power series
　 6.3.1 functional series
　6.3.2 power series and their convergence
　 6.3.3 operations of power series
　exercises 6.3
6.4 expansion of functions in power series
　 6.4.1 taylor and maclaurin series
　6.4.2 expansion of functions in power series
　 6.4.3 applications of power series expansion of
functions
　exercises 6.4
6.5 fourier series
　 6.5.1 orthogonality of the system of trigonometric
functions
　6.5.2 fourier series
　 6.5.3 convergence of fourier series
　6.5.4 sine and cosine series
　 exercises 6.5
6.6 fourier series of other forms
　 6.6.1 fourier expansions of periodic functions with period
2l
　6.6.2 complex form of fourier series
　 exercises 6.6
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