MATH UN1202 Calculus IV
Section 004 Spring 2017

Time and place: TR 11:40--12:55, location 312 Mathematics.
Instructor: Robert Friedman (x4-4355). Office: 605 Mathematics.
Office hours: My office hours are Mondays and Wednesdays, 2--3 PM in 605 Math, but feel free to drop by at any time.
Email: rf@math.columbia.edu
Teaching Assistants: Jingwei Xiao jx2237@columbia.edu, Hidy Han yh2635@columbia.edu, Garrett Kaighn grk2114@columbia.edu. Office hours Jingwei Mondays 9--12, Hidy Thursdays 4--6, Garrett Wednesdays 4--6, all in the Columbia Help Room 406 Mathematics.

This is a fourth semester calculus course. Prerequisites are Calculus I -- III or the equivalent. In particular, you should be familiar with standard techniques of differentiation and integration and with the basic properties of functions of several variables, including partial derivatives and the gradient. The last part of the course assumes some familiarity with complex numbers (although these will be reviewed) as well as Taylor series. The course begins with integration of functions of two and three variables. Next, we study the calculus of vector fields: the various differential operators (grad, curl, div) that can be applied to a function or vector field, types of integrals of vector fields (line integrals, surface integrals), and the fundamental theorems (Green, Stokes, divergence or Gauss theorem) relating differentiation and integration of vector fields. The last part of the course is an introduction to the theory of functions of a complex variable. This theory is important in many applications of mathematics, physics, and engineering, and draws upon the material of the first two thirds of the course.

Text: James Stewart, Calculus: Early Transcendentals, 8th Edition. WebAssign will NOT be used in this course. For more information about various purchasing options, please consult the section on the course textbook in

Homework: There will be weekly problem sets, due at the beginning of class on Tuesdays. Problem sets will consist of problems taken from the textbook or handouts, to be written up and handed in. Homework will be collected in class on Tuesdays, or you can put it in the box on the fourth floor marked Calculus IV with my name by 5 PM of the day it is due. Graded homework will be available outside my office (605 Mathematics).

The first problem set will be due on Tuesday, January 24. You should attempt every homework problem and eventually understand how to do every problem correctly. Collaboration and discussion with your classmates is encouraged, but you must write up assignments individually.

Exams: There will be two 75-minute midterm exams and a final.

• Midterm Exam 1: Thursday, Feb. 16, in class
• Midterm Exam 2: Thursday, March 30, in class (tentative date)
• Final: this is centrally scheduled by the University. The tentative time is Thursday, May 11, 2013, 4:10--7:00 PM, in the usual classroom. It must be taken at the scheduled time.
If you have two final examinations scheduled at the same time, it is the responsibility of the other department to provide an alternate exam. Examinations will not be rescheduled because of travel arrangements -- it is your responsibility to schedule travel appropriately. Makeup midterms will be given only under exceptional circumstances and you will need a note from a doctor or a dean.

Homework: 20%;
Midterm exams: 20% each;
Final exam: 40%.

Help: My office hours are Mondays and Wednesdays, 2--3 PM in 605 Math, and you should always feel free to make an appointment or just drop by. Help is also available without appointment in the Columbia Help Room (406 Mathematics) whenever it is open.

Academic Dishonesty: The vast majority of students do not cheat. Anyone who does so devalues the hard work of the rest of the class and creates a bad atmosphere for all. Anyone found to have cheated on an exam will receive a failing grade for the course and be subject to administrative discipline. If you are struggling with the material or have a problem about an upcoming exam, please discuss it with me instead of resorting to cheating.

Important dates:

January 17: First day of class
February 16: Midterm exam 1
February 21: Drop date (most schools)
March 13--17: Spring break
March 30: Midterm exam 2
April 27: Last day of class
May 11: Final exam (tentative)

### Homework:

This schedule is tentative and may be modified as necessary. Please check here for each week's reading and homework.

 Date Reading Homework Jan. 17, 19 15.1, 15.2: Double integrals. HW#1 due Jan. 24: 15.1: 2, 4, 20, 22, 24, 32, 34, 42; 15.2: 2, 4, 16, 18, 28, 32, 46. Jan. 24, 26 15.3, 15.4, 15.6: Polar coordinates, applications, triple integrals. HW#2 due Jan. 31: 15.3: 6, 10, 22, 26, 28; 15.6: 4, 6, 16, 32, 34, 36. Jan. 31, Feb. 2 15.7, 15.8: Cylindrical and spherical coordinates. HW#3 due Feb. 7: 15.7: 18, 22, 24, 30; 15.8: 8, 10, 22, 24, 30, 42, 48. Feb. 7, 9 15.8, 15.9 Change of variable, snow day. HW#4 due Feb. 16: 15.9: 2, 4, 8, 12, 16. Feb. 14, 16 Review, First Midterm Feb. 16. No homework due. Feb. 21, 23 16.1, 16.2: Vector fields, line integrals. HW#5 due Feb. 28: 16.1: 6, 8, 11-14, 15-18, 24, 26; 16.2: 2, 6, 8, 12, 14, 32, 40. Feb. 28, March 2 16.3, 16.4, 16.5: Fundamental theorem for line integrals, Green's theorem, curl and divergence. HW#6 due March 7: 16.3: 4, 6, 8, 12, 16, 18; 16.4: 4, 6, 12; 16.5: 2, 6, 12, 14, 16, 20, 25, 26, 30, 31, 32 (hint: use 25 and 31(a)). March 7, 9 16.6, 16.7: Parametric surfaces, surface area, surface integrals. HW#7 due March 21: 16.6: 4, 6, 20, 24, 34, 36, 40, 48, 50; 16.7: 14, 20, 24, 26, 30, 44, 46. March 21, 23 16.8, 16.9: Stokes' theorem, divergence theorem. HW#8 due March 28: 16.8: 2, 4, 8, 10, 14, 16, 18; 16.9: 2, 4, 6, 8, 12, 24, 26. March 28, March 30 Review, Second Midterm March 30. No homework due; review complex numbers (Appendix H) and do the problems 2, 4, 6, 10, 12, 26, 30 if needed. April 4, 6 Complex functions 1.   Complex Functions 1 HW#9 due April 11: Problems in Complex Functions 1. April 11, 13 Complex functions 2: Cauchy-Riemann Equation.   Complex Functions 2 HW#10 due April 18: Problems in Complex Functions 2. April 18, 20 Complex functions 3: Contour Integrals and Cauchy's Theorem.   Complex Functions 3 HW#11 due April 25: Problems in Complex Functions 3. April 25, April 27 Review HW#12 due May 2: Review problems. May 2 Optional review session May 11, 4:10 PM FINAL EXAM