Department of Mathematics

Patricia Bauman

Math 442 Spring 2018

Professor: Patricia Bauman

Office: Math 718
Phone: 494-1945
Office Hours: Monday at 1:30-2:30 pm, Wednesday at 2:00-3:00 pm, or by appointment.
Course Lectures:
MWF 10:30-11:20 in REC 309.

Textbook: The Elements of Real Analysis, Second Edition, by Robert G. Bartle, John Wiley and Sons, 1975.

Syllabus: Most of Chapters VI-VIII in the text, and additional topics.

Prerequisite: MATH 44000H and some knowledge of linear algebra (pertaining to matrix multiplication, determinants and inverses of matrices).

Homework will be collected weekly in class on Wednesdays (with some exceptions). No late homeworks will be accepted; however, the lowest homework score will be dropped. The assignments are posted below.

Exams: There will be two midterm exams (in class) and a final exam.
Exam 1 is on Friday, Feb. 23, in class.
Exam 2 is on Wednesday, Apr. 11, in class (on the material covered this semester in Sections 41, 24, and 26).

Course Grade: Your course grade will be computed by the scheme:
Final Score = 25% Homework + 20% Exam 1 + 20% Exam 2 + 35% Final Exam.

Topics Covered:
Mon, Jan. 8: Riemann Integrability of functions of bounded variation on [a,b], Fundamental Theorem of Calculus.
Wed, Jan. 10: Sec. 34. Absolute and conditional convergence of Series in R^p, Cauchy Criterion, nonnegative series, examples.
Fri, Jan. 12: Harmonic series, rearrangement theorem.
Mon, Jan. 15 No class- Martin Luther King Day
Wed, Jan. 17: Sec. 35. Comparison test, limit comparison test for series. Root tests for absolute convergence of series in R^p.
Fri, Jan. 19: Ratio tests and Raabe's tests for absolute convergence of series in R^p. Integral test.
Mon, Jan. 22: Sec. 36. Abel's lemma, Dirichlet's test.
Wed, Jan. 24: Abel's test, application of Dirichlet's test, other examples.
Fri, Jan. 26: Sec. 18. The limit superior and limit inferior. Definition, examples, and equivalent statements.
Mon, Jan. 29: Proof of equivalent statements to the definition of the limit superior and lim inferior.
Wed, Jan. 31: Further results on lim sup and lim inf. Sec. 39. Differentiation of functions from domains in R^p into R^q.
Fri, Feb. 2: Directional derivatives for functions from domains in R^p to R^q. Computation of Derivatives for differentiable functions from domains in R^p to R.
Mon, Feb. 5: Computation of derivatives for differentiable functions from domains in R^p to R^q.
Wed, Feb. 7: Sufficient conditions for a function with domain in R^p and range in R^q to be differentiable.
Fri, Feb. 9: Sec. 40: Differentiability of the product of a scalar-valued differentiable function and a vector valued differentiable function with domain in R^p.
Mon, Feb. 12: The chain rule for differentiability of the composition of a differentiable function f defined on a neighborhood of a point c in R^p with range in R^q and a differentible function defined on a neighborhood of f(c) in R^q with range in R^r.
Wed, Feb. 14: Mean Value Theorems for functions with domain in R^p and range in R or R^q. Interchange of derivatives.
Fri, Feb. 16: Taylor's Theorem for a function of several variables.
Mon, Feb. 19: Sec. 41: Mapping theorems, norms of linear maps.
Wed, Feb. 21: Injective mapping theorem.
Fri, Feb. 23: Exam 1 on Sections 18,34,35,36,39,40.
Mon, Feb. 26: Finished the proof of the injective mapping theorem. Surjective mapping theorem.
Wed, Feb. 28: Finished the proof of the surjective mapping theorem. Open mapping theorem.
Fri, Mar. 2: Inverse mapping theorem.
Mon, Mar. 5: Finished the proof of the inverse mapping theorem.
Wed, Mar. 7: Discussion of exam 1 solutions.
Fri, Mar. 9: Statement and proof of the implicit function theorem for a mapping F(x,y) defined on a neighborhood in R^p x R^q into R^q.
Mon, Mar. 19: Finished the proof of the implicit function theorem.
Wed, Mar. 21: Examples: applications of the implicit function theorem.
Fri, Mar. 23: Section 24: Bernstein approximation theorem.
Mon, Mar. 26: Section 26: Stone approximation theorem
Wed, Mar. 28: Finished the proof of the Stone approximation theorem
Fri, Mar. 30: Stone-Weierstrass theorem.
Mon, Apr. 2: Arzela-Ascoli theorem.
Wed. Apr. 4: Proof of the Arzela-Ascoli theorem.
Fri, Apr. 6: Finished the proof and gave an application of the Arzela-Ascoli Therorem.
Mon, Apr. 9: Metric spaces.
Wed, Apr. 11: Exam 2
Fri, Apr. 13: Contraction mapping theorem in a complete metric space.
Mon, Apr. 16: Application of the Contraction Mapping theorem: Existence-Uniqueness theorem for first-order systems of ode's.
Wed, Apr. 18: Section 43: The Riemann integral in R^p. Sets of ontent zero.
Fri, Apr. 20: Cauchy Criterion for Riemann integrability on cells in R^p. The Riemann integral on other subsets of R^p. Properties of the integral.
Mon, Apr. 23: Discussion of exam 2.
Wed, Apr. 25: Properties of the integral. The integrability theorem.
Fri, Apr. 27: Section 44: Sets with content. Theorem 44.8: Integrability of functions that are bounded and continuous on a set A in R^p assuming that A has content.

Homework Assignments:
1.) Due Wed, Jan. 17: 34H,34K,34N.
2.) Due Wed, Jan. 31: 35D(d,e,f),35I,35J, 36A,36B.
3.) Due Wed, Feb. 7: 35E, 39D,39F.
4.) Due Wed, Feb. 14: 39A,39B,29H,39I,39J(a).
5.) Due Wed, Feb. 21: 40A,40B,40L,40U.
6.) Due Wed, Mar. 7: 41G,41H,41I.
7.) Due Fri., Mar. 23: 41J,K,N. Note: in 41N.(a) you are solving f(x,y,z)=(0,0) for (x,y) as a function of z.
8.) Due Wed, Mar. 28: 24R,24S.
9.) Due Wed, Apr. 4: 26B,26C,26D,26E,26F.
10.) Due Wed, Apr. 18: 8T,8U
11.) Due Wed, Apr. 23: 43L,43M

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