CALCULUS IV
1. General Information
Course: Calculus IV-003, Fall 2011
Prerequisites: Calculus I and III or equivalent
Time: TTh 4:10pm - 5:25pm
Location: 517 HAM Building
Instructor: Sabin Cautis
Office: 624 Mathematics Building
Office Hours: Tuesday and Thursday 5:30-7:00pm (tentative) or after class
E-mail: scautis@math.columbia.edu
Website: includes syllabus, course schedule and homework assignments
http://www.math.columbia.edu/∼scautis/calcIV03/
Homework solutions and grades will be posted on courseworks.
• Text: Calculus, Early Transcendentals by James Stewart (6th edition)
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Homework: Homework will be assigned every week on Tuesday and due the following Tuesday at
the end of class. Assignments will be available on the course homepage. You may get help from any
source but you are strongly encouraged to attempt the problems yourself before seeking assistance.
Any written work submitted must be your own. Your submitted work must be stapled together and
should have your name on every page. Unstapled pages with no name will be discarded.
Homework is to be dropped off in the collection box outside office 410 Math Building until 6PM on
the day it is due, after that homework will be late. Late homeworks will not be accepted for credit.
WARNING: I teach 2 sections so make sure you use the correct box (section 3).
Tests: There will be two midterm exams and a final exam. The tests will be closed book-closed
notes tests. Calculators will not be allowed. The (tentative) dates for the two midterms will be:
Thursday October 6 and Thursday November 17th. The exams will be held in class during the usual
meeting time.
Grades: Grades will be computed as follows: Homework: 20 % (the lowest homework score will be
dropped), Midterms 20% each, Final 40%.
Help Rooms: 406 Math building – the schedule can be found here:
http://www.math.columbia.edu/department/mathhelp.shtml
Final Exam: The final exam schedule will be out shortly. Book your travels accordingly!
Synopsis: The usual fundamental theorem of calculus (which we all should be very familiar with
by now) is a relationship between the integral of df /dx over a region (in this case the region is an
interval) and the integral of f over the boundary of the region (which in this case is two endpoints and
this amounts to evaluating f at these two points).
This theorem has analogues in every dimension. In dimensions 3 these go by the name of the
divergence and Stokes’ theorem. They are quite fundamental and have important applications. For
example, basic results in electrodynamics are essentially applications of these theorems.
The aim of this course is to get to the point of understanding the statements and contents of these
theorems and how to apply them.
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CALCULUS IV
Other Advice Unlike a novel, reading mathematics is a slow process. One could very well spend
hours reading one page. So read it carefully until you understand. Be patient!
Getting help. If you’re having trouble, get help immediately. Everyone who works seriously on
mathematics struggles. But if you don’t get help promptly you will soon be completely lost. The first
places to look for help are my office hours and the course TA in the help room. Talking to your other
classmates can also be very helpful.
Teach to learn! The best way to learn mathematics is to explain it to someone. You’ll find that,
particularly in office hours, I’ll try to get you to explain the ideas. You should also try explaining the
material to each other. The person doing the explaining will generally learn more than the explainee.
2. Schedule
Very rough course schedule, subject to later adjustments:
• Sep 6/8 sect. 15.1, 15.2, 15.3:
Double integrals, Iterated integrals, General regions. Short review.
• Sep 13/15 sect. 15.4, 15.5:
Polar coordinates. Applications of double integrals.
• Sep 20/22 sect. 15.6, 15.7:
Triple integrals. Cylindrical coordinates.
• Sep 27/29 sect. 15.8, 15.9:
Spherical coordinates, Change of variable
• Oct 4 Review.
• Oct 6 MIDTERM I
• Oct 11/13 sect. 16.1, 16.2:
Vector Fields. Line Integrals.
• Oct 18/20 sect. 16.2, 16.3:
Line Integrals. Fundamental theorem for line integrals.
• Oct 25/27 sect. 16.4, 16.5:
Green’s theorem. Curl and divergence.
• Nov 1/3 sect. 16.6, 16.7, 16.8:
Parametric surfaces. Surface integrals. Stokes Theorem.
• Nov 8 Academic Holiday
• Nov 10 sec. 16.9:
Divergence theorem.
• Nov 15 Review.
• Nov 17 MIDTERM II
• Nov 22
TBD.
• Nov 24
Thanksgiving.
• Nov 29/ Dec 1 Notes and Appendix H:
Complex Functions 1 and 2.
• Dec 6/8 Notes:
Complex Functions 3. Course review.