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**Course Code**: B11005Y-B01

**Course Name**: Calculus III-B

**Credits**: 4.0

**Level**: Undergraduate

**Pre-requisite**: Calculus I and II

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

**Instructors**: Null

**Course Description**

This course 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 of this course contains more theoretical analysis which is suitable for students with major other than mathematics. This course is based on Calculus II-B.

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 learn basic theories and methods for differential equation and know fundamental theory for single valued complex function. The course is aimed at laying the foundation for future courses.

**Topics and Schedule **

**Part I**

- Multiple integration (2 weeks)

1.1. The definition of double-integral of functions

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

1.3. Green formula, curve integrals and path independent

1.4. Change of variables of double-integral of functions

1.5. Multiple integral

1.6. Change of variables of multiple integral

1.7. Improper double-integral of functions

- Surface integrals (3 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

- Differential of vector function (4 weeks)

3.1.N dimensional Euclidean space and vector function

3.2.Differential of vector function

3.3.Inverse function and implicit function theorem

**Part II**

- Differential equation (4 weeks)

4.1.Differential equations of a particle in the space

4.2.Normal one order linear differential equation

4.3.High order linear differential equation

4.4.Normal one order differential equation

4.5.Differential simultaneous equations and high order differential equation

4.6.The method of undetermined coefficients for integral computation

- Calculus of variations (2 weeks)

5.1.Function and its extreme value

5.2.Functional extreme and its necessary condition

5.3.Extension

5.4.Problems with additional conditions and Legendre multipliers

- Single valued complex function (4 weeks)

6.1.Complex function shoed by power function

6.2.The basis for normal theory of single valued complex function

6.3.Integral of Analytic Functions

6.4.Cauchy formula and its application

6.5.Residue theorem and its application to complex integral

6.6.Multi-variable function and analytic expansion

**Grading**

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

**Textbook**

[1]School of mathematical sciences, East China Normal university, Mathematical analysis(4^{th} edition), higher education Press, 2010

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

**References**

[1] Gregory Fikhhtengol'ts, A Course of Differential and Integral Calculus, higher education Press, 2005 (Chinese version)

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