## Math V1202: Calculus IV

### Spring 2008

Time and place: MW 11:00--12:15, in 203 Mathematics.
Instructor: Robert Friedman (x4-4355, Office 605 Mathematics).
Office hours: Tentatively Tuesday and Thursday 11--12 AM in 605 Math, but feel free to drop by at any time.
Email: rf@math.columbia.edu
Teaching Assistants: Irena Penev ipenev@math.columbia.edu and Eric Potash elp2109@columbia.edu. Office hours: Irena Monday 11:30-12:30, Thursday 11-12, Friday 11-12; Eric Tuesdays 1-2 PM and 5-6 PM, both in the Math Help Room.

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. The last part of the course assumes some familiar 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) 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.

Page down for [ Homework ]

Text: James Stewart, Calculus: Early Transcendentals, sixth edition, Thomson Brooks Cole, 2008. Available at the University bookstore.

Homework: There will be weekly problem sets, due at the beginning of class on Mondays. The first problem set will be due on Monday, January 28. 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. Calculus IV is much more demanding than previous calculus classes. Students who resort to copying their homework from their classmates, a solution manual or the Web almost invariably come to grief on the exams.

Graded homework will be available on the table outside 605 Mathematics. No late homework will be accepted.

Quizzes: There will be five in class quizzes, of which the lowest will be dropped. The tentative dates for the quizzes are February 4, February 11, March 10, March 31, April 28 (all dates Mondays).

Exams: There will be two 75-minute midterm exams and a final. The tentative dates for the midterms are as follows:

• First Exam, Wednesday, February 20, in class
• Second Exam, Monday, April 7, in class
• Final: this is centrally scheduled by the University. The tentative date and time are May 12, 9 AM--12 Noon, in 203 Mathematics. 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: 10%
Quizzes: 10%
Midterm exams each: 20%
Final exam: 40%.

Help: My office hours are tentatively Tuesday and Thursday 11--12 in, and you should always feel free to make an appointment or just drop by. Help is also available without appointment in the Mathematics Help Room (406 Mathematics) whenever it is open, tentatively 10am--7pm Monday-Thursday, 12:30pm-5pm Friday.The Help Room is staffed both by faculty and teaching assistants, who will be able to help you with questions related to this course.

Important dates:

January 23: First day of class
February 20: First midterm exam
February 26: Drop date
March 17--21: Spring break
April 7: Second midterm exam
May 5: Last day of class
May 12: Final exam

### 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. 23 15.1-15.2: Double integrals. HW#1 due Jan. 28:   15.1: 1, 2  15.2: 4, 10, 12, 20, 22, 26, 30 Jan. 28, 30 15.3-15.4: General regions; Polar coordinates. HW#2 due Feb. 4:   15.3: 2, 6, 12, 18, 20, 28, 48  15.4: 6, 8, 16, 28, 36 Feb. 4, 6 15.5-15.6: Applications; Triple integrals.   Quiz 1 February 4. HW#3 due Feb. 11:   15.5: 6, 12, 16, 24  15.6: 2, 4, 12, 16, 22, 36 Feb. 11, 13 15.7-15.9: Cylindrical and spherical coordinates; Change of variable.   Quiz 2 February 11. HW#4 due Feb. 18:   15.7: 16, 18, 22  15.8: 8, 10, 12, 18, 26, 44  15.9: 2, 6, 20, 24 Feb. 18, 20 Review; First Midterm Feb. 20. HW#5 due Feb. 25:   None Feb. 25, 27 16.1-16.2: Vector Fields; Line Integrals. HW#6 due March 3:   16.1: 2, 6, 10, 11--14, 15--18, 24, 26  16.2: 2, 6, 8, 12, 20, 28, 40 March 3, 5 16.3-16.4: Fundamental theorem for line integrals; Green's theorem. HW#7 due March 10:   16.3: 4, 6, 8, 12, 16, 18  16.4: 2, 4, 6, 14, 18 March 10, 12 16.5-16.6: Curl and divergence; Parametric surfaces.   Quiz 3 March 10. HW#8 due March 24:   16.5: 2, 4, 14, 16, 26, 32  16.6: 4, 6, 22, 24, 40, 44, 46 March 24, 26 16.7-16.8: Surface integrals; Stoke's theorem. HW#9 due March 31:   16.7: 10, 14, 20, 24, 40, 44  16.8: 2, 4, 6, 10, 16, 18 March 31, April 2 16.9: Divergence theorem; Review.   Quiz 4 March 31. HW#10 due April 7:   16.9: 2, 4, 12, 14, 18, 24, 26 April 7, 9 Second Midterm April 7; Complex Numbers. HW#11 due April 14:   Appendix H: 2, 6, 10, 12, 26, 30, 38, 44, 46 April 14, 16 Complex Functions 1.   Complex Functions 1 HW#12 due April 21:    Problems in Handout 1. April 21, 23 Complex Functions 2: Cauchy-Riemann Equation.   Complex Functions 2 HW#13 due April 28:    Problems in Handout 2. April 28, 30 Complex Functions 3: Contour Integrals and Cauchy's Theorem.   Complex Functions 3   Quiz 5 April 28. HW#14 due May 5:    Problems in Handout 3. May 5 Review. May 12, 9am FINAL EXAM