SELECTED TOPICS IN

ADVANCED CALCULUS II

Ma 621

1. Course Description

This course studies: vector, multiple integrals, curves and surfaces, theory of integration, and infinite and power series.

Pre-requisite: MA 620 Selected Topics in Advances Calculus I

2. Goals of the Course

1. To provide a foundation for the calculus that is both rigorous and significant to teachers of secondary-school mathematics.

2. To provide insight into the fundamental concepts of the calculus. (e.g. sequence, limit, continuity, derivative, integral, etc.)

3. To extend and apply further the concepts of undergraduate calculus.

3. Instructional Procedures

1. Lecture/discussion

2. Small group and independent study

3. Use of computer software and graphing calculators

4. Course Content

1. Elements of Partial Differentiation

a. Partial Derivatives

b. Implicit Functions

c. Geometrical Significance of Partial Derivatives

d. Maxima and Minima

e. Composite Functions and the Chain Rule

f. Second Derivatives by the Chain Rule

g. Homogeneous Functions and Euler’s Theorem

h. Derivatives of Implicit Functions

i. Extremal Problems with Constraints

j. Lagrange’s Method

k. Sufficient Conditions for Differentiability

2. Implicit-Function Theorems

a. Implicit Functions

b. Fundamental Theorem and its Generalizations

3. Inverse Function Theorem

a. Inverse Function Theorem in Two Dimensions

b. Mappings and Successive Mappings

c. Transformations of Co-ordinates

d. Curvilinear Co-ordinates

e. Identical Vanishing of the Jacobian

f. Functional Dependence

4. Line Integrals

a. Line Integrals

b. Green’s Theorem

c. Transformations of Double Integrals

d. Exact Differentials

e. Line Integrals Independent of Path

5. Uniform Continuity

6. Infinite Series

7. Uniform Convergence

8. Theory of Integration

5. Evaluation Measures

1. Homework

2. Written Examinations

6. Bibliography

A. Required Text

Taylor and Mann, Advanced Calculus, John Wiley & Sons, Inc., 1983

Note: In mathematics courses it is usually preferable to have a designated textbook which helps to focus the discussion and standardize the language and symbolism.

B. Additional Required Readings

None

C. Supporting Bibliography

Kaplan, W., Advanced Calculus, Addison-Wesley, 1991

Mathematical Association of America, Resources For Calculus Collection, Vol. 2: Calculus Problems for a New Century, M.A.A., 1993

Olmstead, John M.H., Advanced Calculus, Prentice Hall, 1961

Simmons, George, Calculus Gems, McGraw-Hill, 1992

Spiegel, Murray R., Schaum’s Outline Series; Theory and Problems of Advanced Calculus, McGraw Hill, 1968

D. Relevant Periodical Sources

None

E. Relevant Software

Derive

F. Other

TI-82 or TI-83 Graphing Calculator