Syllabus_Calculus I (A)

Course Code: B11002Y-A01

Course Name: Calculus I (A)

Credits: 4.0

Level: Undergraduate

Pre-requisite: Not required

Lecture Time: 80 hrs


Course Description

This course is one of the most important and basic courses for all majors in mathematics. Its basic contents include limit, continuity, derivative and their applications, indefinite integral,definite integral, progression, Fourier series, differentialcalculus of multivariable function,multiple integral, curvilinear integral, surface integral, parametric integral and so on. This course lays the foundation for learning future courses such as differential equation, differential geometry, complex function, real variable function, probability theory, basic physics and rational mechanics.Calculus is also an important course tocultivate basic abilitiesand ways of thinkingwhen students are learning college mathematics. It is crucial for future study and research to master the basis of calculus.This course is divided into three semesters. There are 72 academic hours for teaching and 36 academichours for exercise in the first two semesters. In the last semester, there are 56 and 28 academic hours for each part.

Topics and Schedule

  1. Real number

1.1.Special classes of real numbers and their basic properties

1.2.Dedekind cut and the principle of exact bounds

1.3.Several fundamental lemmas of real number completeness

  1. Limit

2.1.Limit of a sequence

2.2.Properties of a convergent sequence

2.3.Cauchy convergence principle

2.4.Limit of a function

  1. Continuous functions

3.1.Definition of a continuous function

3.2.Properties of a continuous function

3.3.Continuity of the elementary functions

  1. Derivative

4.1.Concept of derivative

4.2.Derivation rules

4.3.Derivative of a parameter-variable function

4.4.Higher derivative

  1. Differential

   5.1. Differentiability and differential

   5.2. Fundamental theorems of differential calculus

   5.3. Differential and functional properties

   5.4. Applications of differentiation in natural science

  1. Indefinite integrals   

   6.1.Concept of indefinite integrals and basic integral formulas

   6.2. Integration by substitution and parts

   6.3. The indefinite integral of a rational function or the function which can be transformed into a rational function

  1. Definite integrals

   7.1. Concept of a definite integral

   7.2. The Newton-Leibniz formula

   7.3. Conditions of integrability

   7.4. Properties of a definite integral

   7.5. Fundamental theorems of calculus

   7.6. Applications of a definite integral

  1. Improper integrals    

   8.1. Concept of an improper integral

   8.2. Properties and convergencecriteria of an infinite integral

   8.3. Improper integralswith one singular point or more

  1. Numerical series

   9.1. Convergence of series

   9.2. Positive series

   9.3. General series

  1. Function sequence and function series

   10.1. Uniform convergence

   10.2. Properties of the uniform-convergence function sequence and function series

  1. P

   11.1. Power series

   11.2. Power series expansion of a function

   11.3. Exponential function of a complex variable, i.e.Euler formula

  1. Fourier series

   12.1. Fourier series

   12.2.Expansion of a  periodic function

   12.3. Proof of convergence theorems

  1.  Limit and continuity of a multivariate function

13.1. Planar point set and multivariate functions

13.2. Limit of a binary function

13.3. Continuity of a binary function  

  1. Differential calculusmultivariate functions

14.1. Differentiability

14.2. Differential method of composite functions

14.3. Directional derivative and gradient

14.4. Taylor formula and the extremum problem

  1. Implicit functions’ theorems and their applications

15.1. Implicit functions

15.2. Implicit function group

15.3. Geometric applications

15.4. Conditional extremum

  1.  Integrals containing a parameter

16.1. Normal integrals containing a parameter

16.2. Abnormal integrals containing a parameter

16.3. Euler integral

  1.  Curvilinear integrals

17.1. First kind ofcurvilinearintegrals

17.2. Second kind of curvilinear integrals

  1.  Multiple integrals

18.1. Concept of double integrals

18.2. Calculation of double integrals in a rectangular coordinate system

18.3. The Green formula and the independence of curvilinear integral and routes

18.4. Variable transformation of double integrals

18.5. Multiple integrals

18.6. Variable transformation formula of multiple integrals

18.7. Improper multiple integrals

  1. Surface integrals

19.1. The first-type surface integrals

19.2. The second-type surface integrals

19.3. Gauss formula and Stokes formula

  1. Vector analysis and basic field theory

20.1. Vector functions

20.2. Differential operation of vector analysis

20.3. Integral formulas of field theory

20.4. Potential field

  1.  Asymptotic expansion

21.1. Asymptotic formulas and asymptotic series

21.2. Asymptotic integrals


B.A. Zorich, Mathematical Analysis, Higher Education Press, 2005


[1] Department of Mathematics, East China Normal University. Mathematical Analysis(the fourth edition).Higher Education Press, 2010

[2] R. Courant, F. John, Introduction to calculus and mathematicalanalysis. Science Press, 2001