I. Number and title of course

Math 711 - Analysis I

A. Our Graduate student may no longer take Math. 418 for graduate credit.

B. The calculus of several variables is becoming more important in the pure and applied sciences. This has been recognized by C.U.P.M. in their various recommendations.

C. To establish a fundamental course in graduate analysis which will be required in both our M.A. end M.S. in Ed. programs.

D. This course replaces a present course ? Math. 613 Calculus of Several Variables.

This course is the first semester of a unified two-sementer analysis sequence designed to provide the students with the basic techniques and theorems of real and complex analysis, and to demonstrate the interplay between these fields of analysis. Material presented here is basic to any subsequent course in analysis.

A. Functions of several variables

1. Topology of Euclidean n-space

2. Functions and limits

3. Compactness and continuity

4. Uniform continuity

5. Convergence and uniform convergence of sequences of functions

B. Partial differentiation

1. Equality of mixed partial derivatives

2. Composite function and the chain rule

3. The Law of the Mean

4. Implicit Function Theorems

5. Transformation and mappings

C. Integration

1. Riemann-Stieltjes integral

2. Multiple integrals

3. Fubini's Theorem

4. Line integrals Green's theorem in the plane

Apostol, T.M. Mathematical Analysis. Addison-Wesley, Reading, Mass., 1957.

Bartle, R.G. The Elements of Real Analysis. John Wiley & Sons, New York, 1964.

Gelbaum, B.R., and Olmsted, J.M.H. Counter examples in Analysis. Holden-Day, San Franciscto, 1966.

Goldstein, A.A. Constructive Real Analysis. Harper & Row, New York, 1967

Olmsted, J.M.H. Advanced Calculus. Appleton-Century-Crofts, New York, 1961.

Rudin, TT. Principles of Mathematical Analysis, 2nd. ed. McGraw-Hill, New York, 1966.

Rudin, W. Real and Complex Analysis. McGraw-Hill, New York, 1966.

Taylor, Q.E. and Mann, W. R. Advanced Calculus, 2nd ed. Xerox College Publishing, 1972.

Bidder, D. V. Advanced Calculus, 2nd Ed. Prentice-Hall, New Jersey' 1961

Lectures, discussions, assigned readings, problem sets, Student presentations, examinations.

Mathematics 417

3 semester hours

IX. Statement of approval

This course proposal was examined in accord with recommended procedures and was approved by the Mathematics Graduate Faculty on September 20, 1973.

Math 711 - Analysis I - Introduction to graduate; functions of several variables, topology of Euclidean spaces continuity and uniform continuity, convergence and uniform convergence of sequences of functions partial differentiation, chain rule, implicit functions theorems, Rieman-Stieltjes integration, multiple integrals, Fubini's theorem, Line integrals. Prerequisites: Mayh 417.

Ph.D. in mathematics with special interest in analysis.