Calculus Ⅱ-B

Course Code: B11004Y-B02

Course Name: Calculus Ⅱ-B

Credits: 4.0

Level: Undergraduate

Pre-requisite: CalculusI

Lecture Time: 20 weeks, 2 sessions/week, 2 hours/session 80学时

Instructors:Null

Course Description

This course is the next course to Calculus I-B which is a fundamental course for undergraduate students from all majors. The requirements for students in this course are to learn fundamentals and methods of calculus and related knowledge about calculus. The category B contains more theoretical analysis which is suitable for students with major other than mathematics. The basic requirements of this course are that students should have fundamental knowledge of multivariable calculus with good understanding of related definitions and methods. In addition, students should know how to think by using calculus. The course is aimed at laying the foundation for Calculus III and other courses.

Topics and Schedule

  1. Improper integration (1.5 weeks)

1.1.The definition of improper integration

1.2.Infinite integrand convergence detection

1.3.Improper integral and convergence detection

  1. Numeric series (1 week)

2.1.Convergence of series

2.2.Positive series

2.3.Normal series

  1. Series of functions (1 week)

3.1.Uniform convergence

3.2.Uniform convergent functions and the characteristics of series of functions.

  1. Power series (1 week)

4.1.Power series

4.2.Power series of function expansion

4.3.The power function of complex variables Euler formula

  1. Fourier series (1.5 weeks)

5.1.Fourier series

5.2.The function periodic with period 2π expansion

5.3.Proof of convergence theorem

  1. The limits and continuity in multivariable function (1 week)

6.1.Point set and multivariable function

6.2.The limits of two variable function

6.3.The continuity of two variable function

  1. Differential calculus of multivariable function (1.5 week)

7.1.Differential

7.2.Differential characteristics of composite functions

7.3.Directional derivatives and gradient

7.4.Taylor formula and extreme point problem

  1. Implicit function theorem and its application(1 week)

8.1.Implicit function

8.2.Implicit function groups

8.3.Geometric application

8.4.Conditional extreme

  1. Integral with parameter (1 week)

9.1.Proper integration with parameter

9.2.Improper integration with parameter

9.3.Euler integration

  1. Curve integrals (1 week)

10.1. Curve integrals of differential 1-forms

10.2. Curve integrals of differential 2-forms

  1. Multiple integration (2 weeks)

11.1. The definition of double-integral of functions

11.2. The computation of double-integral of functions in Cartesian coordinate system

11.3. Green formula, curve integrals and path independent

11.4. Change of variables of double-integral of functions

11.5. Triple integral

11.6. Applications of multiple integration

11.7. N integral

11.8. Improper double-integral of functions

11.9.The proof of change of variables of multiple integration in normal conditions

  1.  Surface integrals (2 weeks)

12.1.Surface integrals of differential 1-forms

12.2.Surface integrals of differential 2-forms

12.3.Gauss and Tomas formula

12.4.Field theory

  1. Differential of vector function (1.5 weeks)

13.1.N dimensional Euclidean space and vector function

13.2.Differential of vector function

13.3.Inverse function and implicit function theorem

Grading

The grades include midterm examination, final examination and assignments of weekly homework

Textbook

School of mathematical sciences, Huadong Normal university, Mathematical analysis(4th edition), higher education Press, 2010

References

[1]R. Courant, F. John, Introduction to Calculus and mathematical analysis (translators: Honglin Zhang and Minqiang Zhou), Science Press, 2001

[2]B.A. Zorich, Mathematical analysis, higher education press, 2005 (Chinese version)

Course Website

To be added