MATH UN1205 Accelerated Multivariable Calculus
Section 001 Fall 2019

Time and place: MW 2:40--3:55, location 520 Mathematics.
Instructor: Robert Friedman (x4-4355). Office: 605 Mathematics.
Office hours: My office hours are tentatively Tuesdays and Thursdays, 1:30--2:30 PM in 605 Math, but feel free to drop by at any time.
Email: rf@math.columbia.edu
Teaching Assistants: TBA tba@columbia.edu , TBA tba@columbia.edu. Office hours TBA, all in the Columbia Help Room 406 Mathematics.

This is a third semester calculus course, which aims to cover the material of Calculus III and much of Calculus IV in one semester. Prerequisites are Calculus I--II or the equivalent. In particular, you should be familiar with standard techniques of differentiation and integration. The course begins with the study of vectors in two and three dimensions, the dot product and cross product, and equations of lines and planes. Next we turn to vector-valued functions of a single variable, and study their limits, derivatives, and integrals. As an application, we define velocity and acceleration for a particle moving in the plane or in space. Then we consider (scalar) functions of several variables and study limits, continuity, and derivatives for such functions. We develop the notions of partial derivatives, directional derivatives, gradients, critical points, maximum and minimum values, the second derivative test, and the method of Lagrange multipliers. We turn then to 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, and various types of integrals of vector fields (line integrals, surface integrals). The course concludes with the fundamental theorems (Green, Stokes, divergence or Gauss theorem) relating differentiation and integration of vector fields.

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 Mondays. Problem sets will consist of problems taken from the textbook, to be written up and handed in. Homework will be collected in class on Mondays, or you can put it in the box on the fourth floor marked Accelerated Multivariable Calculus 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 Monday, September 9. 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: Monday, October 7, in class
• Midterm Exam 2: Wednesday November 6, in class (tentative date)
• Final: this is centrally scheduled by the University. The tentative time is Wednesday December 18, 2019 1:10--4:00 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 tentatively Tuesdays and Thursdays, 1:30--2:30 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:

September 4: First day of class
October 7: Midterm exam 1
October 8: Drop date (most schools)
November 4, 5: Election break
November 6: Midterm exam 2
November 27--29: Thanksgiving break
December 9: Last day of class
December 18: 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 September 4 12.1, 10.3, 12.2: Cartesian and polar coordinates, vectors. HW#1 due September 9: 12.1: 4, 6, 14, 16, 20, 24, 34; 10.3: 8, 12, 16; 12.2: 4, 8, 18, 22, 24. September 9, 11 12.3, 12.4, 12.5: The dot product, cross products, equations of lines and planes. HW#2 due September 16: 12.3: 2, 6, 8, 16, 20, 24, 28, 40, 42; 12.4: 2, 4, 8, 20, 28, 30, 34; 12.5: 2, 4, 16, 24, 28, 32, 34, 46. September 16, 18 10.1, 13.1, 13.2, 13.3, 13.4: Parametric equations, vector functions, derivatives and integrals of vector functions, velocity, acceleration, and arc length. HW#3 due September 23: 10.1: 6, 8; 13.1: 8, 10, 14, 21--26, 42; 13.2: 4, 10, 18, 22, 36, 42; 13.3: 2, 6; 13.4: 4, 8, 12, 16, 18, 20. September 23, 25 14.1, 14.2, 14.3: Functions of several variables, limits and continuity, partial derivatives. HW#4 due September 30: 14.1: 16, 22, 24, 28, 46, 54, 68, 70; 14.2: 6, 10, 18, 38, 40; 14.3: 16, 22, 24, 28, 54, 62, 76(a)--(d). September 30,   October 2 14.4, 14.5, 14.6: Tangent planes and linear approximation, chain rule, directional derivatives and the gradient. HW#5 due October 7: 14.4: 2, 4, 12, 18, 26, 28; 14.5: 6, 8,12, 22, 28, 32; 14.6: 4, 8, 12, 16, 22, 26, 28, 50, 54. October 7, 9 First Midterm October 7; 14.7: Maximum and minimum values HW#6 due October 14: 14.7: 2, 6, 8, 14, 22, 32, 42, 46, 48. October 14, 16 14.8, 15.1: Lagrange multipliers, double integrals. HW#7 due October 21: 14.8: 4, 8, 12, 18; 15.1: 2, 4, 20, 22, 24, 32, 34, 42. October 21, 23 15.2, 15.3, 15.6: Double integrals, polar coordinates, triple integrals. HW#8 due October 28: ; 15.2: 2, 4, 16, 18, 28, 32, 46; 15.3: 6, 10, 22, 26, 28; 15.6: 4, 6, 16, 32, 34, 36. October 28, 30 15.7, 15.8, 15.9: Cylindrical and spherical coordinates, change of variable. HW#9 due November 6: 15.7: 18, 22, 24, 30; 15.8: 8, 10, 22, 24, 30, 42, 48; 15.9: 2, 4, 8, 12, 16. November 6 Second Midterm November 6. No homework due. November 11, 13 16.1, 16.2, 16.3: Vector fields, line integrals, fundamental theorem of line integrals. HW#10 due November 18: 16.1: 6, 8, 11-14, 15-18, 24, 26; 16.2: 2, 6, 8, 12, 14, 32, 40; 16.3: 4, 6, 8, 12, 16, 18. November 18, 20 16.4, 16.5, 16.6,: Green's theorem, curl and divergence, parametric surfaces. HW#11 due November 25: 16.4: 4, 6, 10, 12; 16.5: 2, 6, 12, 14, 16, 20, 25, 26, 30, 31, 32 (hint: use 25 and 31(a)); 16.6: 4, 6, 20, 24, 34, 36. November 25 16.7: Surface area, surface integrals. HW#12 due December 2: 16.6: 40, 48, 50; 16.7: 14, 20, 24, 26, 30, 44, 46. December 2, 4 16.8, 16.9: Stokes' theorem, divergence theorem. HW#13 due December 9: 16.8: 2, 4, 8, 10, 14, 16, 18; 16.9: 2, 4, 6, 8, 12, 24, 26. December 9 Review December 18,   1:10 PM FINAL EXAM