Course Code: B11002Y-A01
Course Name: Calculus I (A)
Credits: 4.0
Level: Undergraduate
Pre-requisite: Not required
Lecture Time: 80 hrs
Instructors:
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
1.1.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
2.1.Limit of a sequence
2.2.Properties of a convergent sequence
2.3.Cauchy convergence principle
2.4.Limit of a function
3.1.Definition of a continuous function
3.2.Properties of a continuous function
3.3.Continuity of the elementary functions
4.1.Concept of derivative
4.2.Derivation rules
4.3.Derivative of a parameter-variable function
4.4.Higher derivative
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
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
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
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
9.1. Convergence of series
9.2. Positive series
9.3. General series
10.1. Uniform convergence
10.2. Properties of the uniform-convergence function sequence and function series
11.1. Power series
11.2. Power series expansion of a function
11.3. Exponential function of a complex variable, i.e.Euler formula
12.1. Fourier series
12.2.Expansion of a 2π periodic function
12.3. Proof of convergence theorems
13.1. Planar point set and multivariate functions
13.2. Limit of a binary function
13.3. Continuity of a binary function
14.1. Differentiability
14.2. Differential method of composite functions
14.3. Directional derivative and gradient
14.4. Taylor formula and the extremum problem
15.1. Implicit functions
15.2. Implicit function group
15.3. Geometric applications
15.4. Conditional extremum
16.1. Normal integrals containing a parameter
16.2. Abnormal integrals containing a parameter
16.3. Euler integral
17.1. First kind ofcurvilinearintegrals
17.2. Second kind of curvilinear 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
19.1. The first-type surface integrals
19.2. The second-type surface integrals
19.3. Gauss formula and Stokes formula
20.1. Vector functions
20.2. Differential operation of vector analysis
20.3. Integral formulas of field theory
20.4. Potential field
21.1. Asymptotic formulas and asymptotic series
21.2. Asymptotic integrals
Textbook
B.A. Zorich, Mathematical Analysis, Higher Education Press, 2005
References
[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
