**Math 447, Real Variables **

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**Instructor:** Prof. Zhong-Jin Ruan

**Classroom: ** 141AH, TuTh 11:00am - 12:20pm

** Office Hour:** TuTh 1-2 pm, or by appointment.

** Office: ** 353 Altgeld Hall

** Email:** z-ruan@illinois.edu

**Web page: ** http://www.math.uiuc.edu/~ruan/447Spring17.html

This course provides a careful development of elementary analysis for those who intend to take graduate courses in mathematics. Topics include completeness property of real number system; basic topological properties of n-dimensional space; convergence of numerical sequences and series of functions; properties of continuous functions; and basic theorems concerning differentiation and Riemann integration.

**Textbook:**
Elementary Analysis: The Theory of Calculus by Kenneth Ross, 2nd edition, 2013.

** Pre-requisite:** Math 242 and Math 347, or equivalent.

** Homework:** Homework will be assigned each week and will be due in class on the following dates:

Part I: Thursdays Jan 26, Feb 2, 9, 16;

Part II: Thursdays Mar 9, 16, 30, Apr 6;

Part III: Tuesdays Apr 25, May 2.

** No late homework will be accepted.** If you have a reasonable excuse for missing an assignment, I will score it by the average of the other assignments.

** Exams:** We will have two midterm exams and a final exam.

** Exam1:** Thursday February 23

** Exam2:** Thursday April 13

** Final Exam:** Thursday May 11, 7-10pm at 141Altgeld Hall.

**Grading policy:** There will be total of 500 points, which can be
computed as follows.

Homework: | 10 x 10 pts | 100 pts |

Exams: | 2 x 100 pts | 200 pts |

Final Exam: | 200 pts | |

Total: | 500 pts |

Your grade will be based on the total scores.

** Homework **
** ********************************************************* **

** HW#1:** #4.5, 4.6, 4.7, 4.14a), 4.16. Due Thursday, January 26, 2017.

** HW#2:** #8.2c),d), 8.4, 8.6a), 8.8a), 8.10. Due Thursday, February 2, 2017.
** Solution ** [pdf]"

** Practice HW (No need to hand in): ** #9.1, 9.2, 9.4, 9.8a), c), 9.15.

** HW#3:** #10.6a), b), 10.7, 10.10, 12.2, 12.4, 12.10. Due Thursday, February 9, 2017.
** Solution ** [pdf]"

** Practice HW (No need to hand in):** 10.1, 10.4, 12.3.

** HW#4:** #11.4 consider (x_n), (z_n); 11.6; 12.6 a), c); 14.2 e), f), g); 14.4a), b); 14.12. Due Thursday February 16, 2017.
** Solution ** [pdf]"
** Practice HW (No need to hand in): ** #11.11, 14.7, 14.10.

** The 1st exam**
will be given on Thursday February 23, 2017 from 11:00am to 12:15pm
** Solution ** [pdf]"

** HW#5:** #13.4, 13.12, 13.13; 17.10a), b), 17.12a), b). Due Thursday March 2, 2017.
** Solution ** [pdf]"
** Practice HW (No need to hand in):** #13.3, 13.9, 13.11; 17.3, 17.9c), d).

** HW#6:** #18.2, 18.4, 18.6, 18.8, 19.2c), 19.6a), b). Due Thursday March 9, 2017.
** Solution ** [pdf]"

** HW#7:** #13.8a), 19.7a), b), 19.10, 21.2, 21.8, 21.10a),d). Due Thursday March 16, 2017.
** Solution ** [pdf]"

**Practice HW (No need to hand in)** #21.1, 21.3, 21.7.

** HW#8:** #22.2, 22.4a),b),c), 22.6a), 22.8, 23.2b),d), 23.6. Due Thursday March 30, 2017.
** Solution ** [pdf]"

(Hint: You may apply 22.3 to 22.4b).)

** Practice HW (No need to hand in). ** #22.1, 22.5, 22.11, 23.1.

** HW#9: ** #24.4, 24.6, 24.14, 25.4, 25.6, 25.10. Due Thursday April 6, 2017

** Practice HW (No need to hand in) ** #24.7, 24.11, 24.13, 25.5, 25.9.

** For section 28 and 29, we only have practice homework (No need to hand in): ** #28.4, 28.6, 28.8, 28.14, 29.10, 29.12, 29.14.

** The 2nd exam** will be given on Thursday April 13, 2017 from 11am to 12:15pm.
** Solution ** [pdf]"

** HW#10: ** #32.2, 32.6, 32.8, 33.4, 33.8a), 33.10. Due Tuesday April 25, 2017.

** Practice HW (No need to hand in) ** #33.3, 33.7

** HW11:** #33.14, 34.2a), 34.4, 34.6, 34.7, 34.8a). Due Tuesday May 2, 2017.