up at Math 125B: Real Analysis

UC Davis Math 125B
Real Analysis

Basic information

CRN code: 52679
Room/Time of the Lectures: 9:00-9:50a.m. MWF in Roessler 55
Discussion Section: 9:00-9:50a.m. Thursday in Roessler 55
Instructor: Professor Alexander Soshnikov
Office: 3140 Mathematical Sciences Building
Office Hours: Tue 4:10-5p.m., Wed 3:10-4p.m, and Thur. 4:10-5p.m.
E-mail: soshniko@math.ucdavis.edu
 
TA : Andrew Port
Office: 2232 Mathematical Sciences Building
Office Hour: Monday 2:10-3p.m.
E-mail: aaport@math.ucdavis.edu
 
TA : Eunghyun Lee
Office: 3229 Mathematical Sciences Building
E-mail: ehnlee@math.ucdavis.edu
 
Midterm : May 7th (Wednesday), in class.
Final: June 9th (Monday), 6:00-8:00p.m. in Roessler 55.
Grading: Problem sets 20%, Midterm 30%, Final 50%
Webpage: http://www.math.ucdavis.edu/~soshniko/125b
 

Textbook:

An Introduction to Analysis by William R. Wade , 3rd edition

Syllabus
If you have a problem with the file, please download the syllabus from http://www.math.ucdavis.edu/courses/syllabi

Homework

Homework will be assigned each Friday and will be due in class one week later.

No late submission allowed except for cases of true emergency.

While the students are allowed (and encouraged!) to discuss homework problems in groups, each student has to write down his/her own homework!

Extra problems (i.e. problems not from the textbook) are part of the hws and WILL contribute to the grade.

Homework 1 (due Friday, April 11 in class):

Reading: Sections 5.1 and 5.2.

Problems from the textbook:

5.1.2, 5.1.4, 5.1.5, 5.1.7, 5.1.8, 5.1.9 (pages 115-117); 5.2.2, 5.2.3, 5.2.5 (page 126).

Extra Problem:

Let f(x) be a function on the closed interval [0,1] defined in the following way. If x is rational, then f(x)=1/q, where x=p/q and (p,q)=1 (i.e. the greatest common divisor of p and q is 1). If x is irrational, then f(x)=0. Prove or disprove that f(x) is integrable on [0,1].

Solutions of HW1

Homework 2 (due Friday, April 18 in class):

Reading: Sections 5.2 and 5.3.

Problems from the textbook:

5.2.6, 5.2.7, 5.2.8 (pages 126-127), 5.3.2, 5.3.3, 5.3.4 (page 134).

Extra Problem 1:

Let f(x) be integrable on the interval [a,b]. Prove or diprove the following statements:

a) g(x)=sin(f(x)) is integrable on [a,b].

b) h(x)=tan(f(x)) is integrable on [a,b].

Extra Problem 2:

Does the statement of the First Mean Value Theorem (Th. 5.24 in the textbook) holds without the condition that g(x) is positive on [a,b] ? Either prove the statement or find a counter-example.

Extra Problem 3:

Let f(x) be an integrable function on the interval [a,b] and F(x) be defined as the integral of f from a to x. Is it true that F(x) is differentiable on [a,b] ? Prove or find a counter-example.

Extra Problem 4:

Prove that the following definition is equivalent to the definition 5.17 (ii) (on page 117) of the convergence of the Riemann sums of f:

Definition.

The Riemann sums of f are said to converge to I(f) as ||P|| go to zero if and only if for any epsilon >0 there exists delta>0 such that for any partition P of the interval [a,b] with the norm less than \delta (i.e. ||P||< delta) and for arbitrary choice of t_j in each interval of the partition [x_{j-1}, x_j] the difference between the Riemann sum and I(f) is less than epsilon in absolute value.

Solutions of HW2

Homework 3 (due Friday, April 25 in class):

Reading: Sections 5.4 and 11.1.

Problems from the textbook:

5.3.8 (page 135), 5.4.1 a),d), 5.4.3, 5.4.4 a),c),e), 5.4.6, 5.4.8, 5.4.9 (pages 141-142), 11.1.1 a),c), 11.1.2 b),c), 11.1.4 (page 329).

Solutions of HW3

Homework 4 (due Friday, May 2 in class):

Reading: Sections 11.1, 11.2, and 8.2.

Problems from the textbook:

11.1.5, 11.1.6, 11.1.8 a), b), 11.1.10 a), b), d), e) (pp. 329-331), 8.2.4, 8.2.5b), 8.2.6b), 8.2.8 (pp. 240-241), 11.2.3, 11.2.4 (p. 338).

Solutions of HW4

Homework 5 (due May 9 in class):

Reading: Sections 11.2 and 11.3

Problems from the textbook:

11.2.7, 11.2.8, 11.2.9, 11.2.10 (pp. 338-339), 11.3.2 a),b), 11.3.4, 11.3.5 b) (pp.347-348).

Solutions of HW5

Homework 6 (due May 16 in class):

Reading: Section 11.4

Problems from the textbook:

11.4.2, 11.4.3, 11.4.4, 11.4.5, 11.4.6, 11.4.7, 11.4.8, 11.4.9, 11.4.10 (pages 350-352).

Solutions of HW6

Homework 7 (due May 23 in class):

Reading: Section 11.5 and 11.6

Problems from the textbook:

11.5.1, 11.5.3 a) and c), 11.5.4, 11.5.6, 11.5.8, 11.5.10 (pages 357-358), 11.6.1 a) and c), 11.6.3, 11.6.4 (page 368).

Solutions of HW7

Homework 8 (due Monday, June 2 in class):

Reading: Section 11.6 and 12.1

Problems from the textbook:

11.6.5, 11.6.6, 11.6.7 (page 368), 12.1.1, 12.1.2, 12.1.5, 12.1.6, 12.1.7 (page 393).

Solutions of HW8

Lectures:

Lecture 1 (March 31): Introduction. Riemann Integral (Section 5.1)

Lecture 2 (April 2): Riemann Integral (Section 5.1).

Lecture 3 (April 4): Riemann Integral (Section 5.1).

Lecture 4 (April 7th): Riemann Sums (Section 5.2).

Lecture 5 (April 9th): Riemann Sums (Section 5.2).

Lecture 6 (April 11th): Fundamental Theorem of Calculus (Section 5.3).

Lecture 7 (April 14th): Fundamental Theorem of Calculus (Section 5.3).

Lecture 8 (April 16th): Improper Integrals (Section 5.4).

Lecture 9 (April 18th): Improper Integrals (Section 5.4).

Lecture 10 (April 21): Multivariate Differentiation. Partial Derivatives (Section 11.1).

Lecture 11 (April 23): Multivariate Differentiation. Partial Derivatives (Section 11.1).

Lecture 12 (April 25): Linear Transformations (Section 8.2).

Lecture 13 (April 28): Definition of Differentiability (Section 11.2).

Lecture 14 (April 30): Definition of Differentiability (Section 11.2).

Lecture 15 (May 2): Relations to Univariate Concepts (Section 11.3).

Lecture 16 (May 4): Preparation for Midterm.

Midterm (May 7).

Lecture 17 (May 9): Chain Rule (Section 11.4).

Lecture 18 (May 12): Chain Rule (Section 11.4).

Lecture 19 (May 14): Mean Value Theorem (Section 11.5).

Lecture 20 (May 16): Taylor's Formula (Section 11.5).

Lecture 21 (May 19): Inverse Function Theorem (Section 11.6).

Lecture 22 (May 21): Inverse Function Theorem (Section 11.6).

Lecture 23 (May 23): The Implicit Function Theorem (Section 11.6).

Lecture 24 (May 28): Integration on R^n. Jordan Regions (Section 12.1).

Lecture 25 (May 30): Integration on R^n. Jordan Regions (Section 12.1).

Lecture 26 (June 2): Riemann Integration on Jordan Regions (Section 12.2)

Lecture 27 (June 4): Riemann Integration on Jordan Regions (Section 12.2)

Final Exam: June 9th (Monday) from 6p.m. till 8p.m. in Roessler 55