SYLLABUS
I. MATH 451-452 Advanced Calculus I, II 3, 3 credit hours
II. 2001-2002 academic year INSTRUCTOR: Dr. Keith Stafford
Office: 103 Farr-Chinnock, ext. 6413
e-mail: kstafford@inetlmu.lmunet.edu
Conference Hours (reserved times to assist students):
Monday – 1:00 p.m. - 2:15 p.m.
Tuesday – 2:15 p.m. - 3:30 p.m.
Wednesday – 1:00 p.m. - 2:15 p.m.
Thursday – 2:15 p.m. - 3:30 p.m.
Friday – by appointment only
III. PREREQUISITE: Math 231-232 or equivalent.
IV. Course description/GOALS: Emphasis is on the rigorous processes of analysis. The limit theorems, properties of continuous functions, existence of the integral, convergence and uniform convergence will be carefully treated. The course includes an introduction to point set topology and abstract spaces. The aim is to present a modern treatment that displays the structure of analysis as a self-contained subject and to enable the student to continue toward research or graduate studies.
V. RELATIONSHIP OF COURSE TO CONTENT AREA KNOWLEDGE AND SKILLS: This course provides opportunities to increase one's ability to reason with abstract concepts at a level beyond that offered in the prerequisite calculus courses. Topological concepts increase understanding of and ability to represent configurations in two- and three-dimensional spaces. The student is presented with proofs of theorems and problems having several steps and is required to communicate mathematics through clear writing and verbalization of proofs to like theorems and problems. Although the student's ability to use the mathematics in modeling may be enhanced, the primary emphasis of the course is on the theory of analysis.
VI. TEXT: Fitzpatrick, Patrick M., Advanced Calculus, PWS Publishing Co., Boston, 1996.
VII. COURSE OBJECTIVES: 1) Become familiar with the language, basic concepts, and standard analytical results. 2) Gain comprehensive awareness of the structure of the real number system. 3) Proceed logically from hypothesis to conclusion, employing abstract reasoning. 4) Acquire the basic topological concepts used in analysis. 5) Work with the derivative and integral from a more advanced perspective than that presented in freshman and sophomore level courses.
VIII. METHODS OF INSTRUCTION AND LEARNING: Lectures and problem solving will constitute the principal method of instruction, with attention given to the nature of various proofs and analysis of theorems. Homework, class discussion, and some recitation and board work will be expected of all students. Students are encouraged seek help pertaining to assignments and course material.
IX. UNITS OF INSTRUCTION / OUTLINE OF COURSE CONTENT:
0. Sets and functions. The field and positivity axioms for the real numbers.
1. The real number system: the Completeness Axiom. The Archimedean Property and density. Some inequalities and algebraic identities.
2. Numerical sequences and convergence of them. Monotone sequences and, the Bolzano-Weierstrass and Nested Interval theorems
3. Continuity and limits of real-valued functions. The Extreme and Intermediate Value theorems. Images and inverses of functions. Uniform continuity.
4. Derivatives and rules for differentiating. The differentiation of compositions and inverses. The Lagrange and Cauchy Mean Value theorems. Leibnitz notation.
5. The natural logarithm and trigonometric functions as solutions to differential equations. Inverses: the exponential and inverse trigonometric functions.
6. Characterization and properties of the real line integral. The first fundamental theorem. The convergence of Riemann and Darboux Sums.
7. The second fundamental theorem and the existence of solutions of differential equations. Two classical integration methods and the approximation of integrals
8. Approximation by and convergence of Taylor polynomials. The Lagrange Remainder theorem, Cauchy Integral Remainder formula and binomial expansion. The Weierstrass approximation.
9. Definition and convergence of series. Pointwise and uniform convergence of function sequences. The uniform limit and power series. An interesting example.
10. The linear structure of
Euclidean n-space (
) and the inner product.
Convergence of sequences.
Interiors, exteriors, and boundaries of subsets.
11. Characterization of continuous
mappings, compactness, and connectedness in 
.
12. Definition of a metric space. Open sets, closed sets, and sequential convergence. Completeness: the Contraction Mapping Principle and existence of nonlinear differential equations. Continuous mappings, compactness, and connectedness.
13. Calculus of multi-variable functions. The Mean Value Theorem and directional derivatives.
14. Approximating real-valued functions and nonlinear mappings, the Inverse Function theorem, and the Implicit Function theorem (as time allows).
15. Integrating functions of several variables. Integrating on a path (line) or surface.
X. REQUIRED READINGS: Sections of the textbook as they are covered in class.
XI. SUGGESTED READINGS/BIBLIOGRAPHY: Appendix A of the course required textbook; Rudin, Principles of Mathematical Analysis, McGraw-Hill.
XII. COURSE REQUIREMENTS / METHOD OF ASSESSMENT AND EVALUATION: Class attendance is required and students are expected to participate in discussion. Attendance may affect grades as follows: One-half (1/2) point is deducted from your final grade percentage for each fifteen (15) minutes of class missed. Collected homework, a midterm, and a final exam are used to evaluate subject knowledge and comprehension. Each of these three graded works, comprising the main measure of grades, is equally weighted. Any of the three assessment forms may contain individual and group work. Weighting of tests and final exam will be at the discretion of the professor, but conform to the following scheme:
88 & up A 67 to 71 B- 46 to 50 D+
83 to 87 A- 62 to 66 C+ 40 to 45 D
78 to 82 B+ 56 to 61 C 35 to 39 D-
72 to 77 B 51 to 55 C- 00 to 34 F
Only one final exam is given on the date fixed by the university, designated in the LMU current schedule of classes, excepting the need for an early grade of a graduating senior.
XIII. DATE OF REVISION: August 20, 2001.