## Prof. Ilya Kofman

Office:   607 Mathematics,  phone: 854-3210
Email:   ikofman@math.columbia.edu
Website:   http://www.math.columbia.edu/~ikofman/

 Course Time and Place: Section 003:   1:10pm - 2:25pm   Monday and Wednesday  in 417 Mathematics Section 004:   2:40pm - 3:55pm   Monday and Wednesday  in 417 Mathematics

Textbook:  James Stewart, Calculus: Early Transcendentals Fifth Edition, 2003. ISBN 0534393217. Available at the University Bookstore or online: AddALL.

Material Covered:  This course covers the "differentiable" part of multivariable calculus. The central part is the study of functions of several variables, partial derivatives, and optimization problems using Lagrange multipliers. We also study vectors, vector-valued functions, parametric curves, and a few special topics (Cramer's rule, complex numbers, 2nd order linear differential equations). Roughly, chapters 12, 13, 14, 17 of our textbook.

Prerequisites:  Calculus I or the equivalent.

Homework:  Assignments will be announced in class and then posted on this website in the column marked "Due". Any changes will be announced in class. Only the ten best homework scores will be counted toward the final grade. Late homework will not be accepted. The listed exercises from the textbook are strongly recommended as practice, but they will not be collected. Answers for these are in the back of the book. I highly recommend working jointly on homework problems with fellow students, but in the end you must hand in your own work.

Grading:  The course grade will be determined as follows: 20% HW + 20% Midterm Exam 1 + 20% Midterm Exam 2 + 40% Final Exam.

Without exception, you must take the final exam at the time scheduled by the university.

Help:  My office hours are Tuesday 1-2pm in my office, 607 Mathematics,  and Tuesday 2-3pm in Barnard help room, 333 Milbank, which is open with people to answer questions all day long on weekdays.

Optimal Method of Study:  (1.) Come to class.  (2.) Read the relevant sections after class.  (3.) Do the homework. Leave time to think--do not put homework off until it is due!  (4.) Compare your solutions with other students to improve what you hand in.   (5.) Come to office hours or the help room with any remaining questions.

Calculators:  Calculators — in particular graphing calculators — are not required for this course. If you have one, you are welcome to use it when you do your homework. However, calculators will not be allowed during any exams.

Written work:  We write to communicate. Please bear this in mind as you complete assignments and take exams. Work must be neat and legible to receive consideration. You must explain your work in order to obtain full credit; an assertion is not an answer. For specific suggestions see A guide to writing in mathematics classes.

Goals:  Make calculations with accuracy, intelligence and flexibility;  explain the basic concepts of calculus clearly and reason logically with them;  solve extended problems, with good judgment in the choice of tools and in checking answers.

## Schedule of Lectures

 Class Topic Read Exercises Due Jan 19 Vectors §12.1, 12.2 p.797: 3, 9, 15, 21; p. 798: 41; p.805: 7, 21 Jan 24 Vectors, dot product §12.2, 12.3 p.806: 31, 37, 39; p.813: 23, 27, 43, 49, 51. Jan 26 Cross product §12.4 p.820: 5, 9, 23; p.821: 33, 41. Jan 31 Lines and planes §12.5 p.830: 9, 13, 27, 33; p.831: 53, 73. problem set 1 Feb 02 Polar coordinates §10.3,10.4 p.677-8: 7, 9, 19, 21, 23, 33; p.683: 1, 3, 5 Feb 07 Surfaces and coordinate systems §12.6,12.7 p.837: 3, 7; p.838: 19, 25, 35; p.843: 41, 55, 65. problem set 2: p.821: 40; p.830: 36, 38, 42 Feb 09 Parametrized curves in space §13.1, 13.2 p.656: 7, 13, 21; p.855: 13, 19-25; p.861: 5, 11. Feb 14 Review SAMPLE MIDTERM 1 problem set 3: p.678: 54; p.683: 32, 46; p.838: sketch 30,32,34,36 Feb 16 Midterm 1 SOLUTIONS SURFACES for #6 Feb 21 Arc length and curvature §13.3 p.868-869: 3, 11, 13, 27, 39 Feb 23 Velocity and acceleration §13.4 p.879: 7, 11, 15, 19, 25 problem set 4: p.856: 34, 40; p.861: 30, 32 Feb 28 Functions of several variables §14.1 p.898: 5, 13; p.899: 30, 35; p.900: 47, 55 Mar 02 Limits, continuity, partial derivatives §14.2, 14.3 p.908: 7, 11; p.919-921: 4, 7, 27, 59, 67 problem set 5: p.868: 12, 20, 42, 49; p.879: 28,34 Mar 07 Tangent planes and linear approximation §14.4 p.930-931, 5, 11, 21, 29, 35 See HW note Mar 09 Chain rule §14.5 p.938-939: 5, 11, 21, 39, 43, 51. p.899: 30,32,56,58; p.908: 14,16; p.920: 56,66,68 Spring Break Mar 21 Review SAMPLE MIDTERM 2 Mar 23 Midterm 2 See HW note problem set 7: p.930: 16,38,42; p. 938: 16,38,44 Mar 28 Gradient, directional derivatives §14.6 p.951-952: 9, 33, 43(a), 57, 61. Mar 30 2nd order derivatives, Local extrema §14.7 p.961-962: 1, 3, 13, 19, 29. Apr 04 Local extrema §14.7 p.962-963: 37, 39, 41, 49, 53. problem set 8: p.952: 34,38,48,52; p.961: 18,44 Apr 06 Lagrange multipliers §14.8 p.970-971: 1, 3, 11, 19, 23. Apr 11 Lagrange multipliers §14.8 p.971: 33, 39, 43. problem set 9: p.961: 14,16; p.971: 18,26,32 Apr 13 Systems of linear equations; Cramer's rule Notes and exercises Apr 18 Complex numbers §Appendix G p.A56: 21, 29, 33, 39, 41. problem set 10: Exercises in Notes; use Cramer's rule to redo p.971: 26 Apr 20 Complex exponential, Euler's formula Complex Analysis, ch.1,2 p.A56: 18, 23, 31, 35, 37 Apr 25 2nd order linear differential equations §17.1,17.2 p.1147: 3, 9, 11, 15, 17, 21, 27, 31, 33 problem set 11: p.A56: 32, 34, 38, 46; Problem A Apr 27 Review May 02 Review Review materials for final exam problem set 12: p.1154: 8, 20a, 22a; p.1168: 8, 12 May 09  (Sec. 3)May 11  (Sec. 4) Final Exam