AdvancedMathematics(Ⅰ)高等数学(上)

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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