Calculus of One and Several Variables

Course Description and Grading Breakdown
We will cover multivariable Calculus, follow Spivak's monograph Calculus on Manifolds. The book's prerequisites are a rigorous course in Calculus (i.e. 1a) and a course in linear algebra (i.e 1b). So it is perfect. The course breaks naturally into two parts. In the first part ( first three chapters of Spivak), we develop the theory
of differentiation and Riemann integration in several variables. In a sense, this is just a natural extension of 1a, but with a greater emphasis on the linear algebra that shows up because we are working in several variables. The second part, which will seem more abstract, and is covered in the last two chapters of Spivak is the study of differential forms. These are things which you integrate on curves and surfaces (and, in general, manifolds.) The main point of this part of the course is to prove all variants of Stokes' theorem (the classical Stokes' theorem, Green's theorem, Gauss' divergence theorem, and others) as a single theorem which can be thought of as the fundamental theorem of calculus. Spivak's book, a masterpiece, is famous for its brevity. This will be great for you. It can be read from cover to cover. This course will be a complete success if, at the end, you understand it from cover to cover.

There will be eight problem sets, due on Monday at 2 A.M. There will be a midterm and final. The grades will be distributed:
Problem sets 40%
Midterm        30%
Final             30%

Course Meeting Time and Location
Monday, Wednesday and Friday
10:00 - 10:55 am 
B122 Gates Chemical Laboratory (GCL, Building #26)

Course Instructor Contact Information and Office Hours
276 Cahill

I will hold office hours on Thursdays and Sundays  7-8 P.M. in room 326 of the Fairchild Library.

TA Contact Information and Office Hours
Pooya Vahidi Ferdowsi 1-J Math Building Office Hours: Fridays 6-8 pm in room 103 of Downs
Josh Frisch  1-I Math Building Office Hours: Sundays 8-9 pm in room 103 of Downs
Lingfei Yi 2-C Math Building Office Hours: Saturdays 7-8 pm in room 103 of Downs

Course Schedule and Textbook

 Date Topic 
April 2  1.1 Norm and Inner Product
 April 4  1.2 Subsets of Euclidean Space
 April 6  1.3 Functions and Continuity
 April 9  2.1 Basic Definitions
 April 11  2.2 Basic Theorems
 April 13  2.3 Partial Derivatives
 April 16  2.4 Derivatives
 April 18  2.5 Inverse Functions
 April 20  2.6 Implicit Functions
 April 23  3.1 Basic Definitions
 April 25  3.2 Measure Zero and Content Zero
 April 27  3.3 Integrable functions
 April 30  3.4 Fubini's Theorem
 May 2  3.4 Fubini's Theorem cont.
 May 4  3.5 Partitions of Unity
 May 7  3.6 Change of Variable
 May 9  3.6 Change of Variable cont.
 May 11  4.1 Algebraic Preliminaries
 May 14  4.1 Algebraic Preliminaries cont.
 May 16  4.2 Fields and Forms
 May 18  4.2 Fields and Forms cont.
 May 21  4.3 Geometric Preliminaries
 May 23  4.4 Fundamental Theorem of Calculus
 May 30  5.1 Manifolds
 June 1  5.2 Fields and Forms on Manifolds
 June 4  5.3 Stokes' theorem on Manifolds
 June 6  5.4 The Volume Element
 June 8  5.5 The Classical Theorems

Course Policies
Late work - No late work accepted without permission in advance.

 Date Posted Assignment  Due Date  Solution 
 4/2/2018 Problem set 1  4/9/2018
 4/9/2018 Problem Set 2  4/16/2018
 4/16/2018 Problem Set 3  4/23/2018
 4/23/2018 Problem Set 4  4/30/2018
 5/7/2018 Problem Set 5  4/30/2018
 5/14/2018 Problem Set 6  5/21/2018
 5/21/2018 Problem Set 7  5/29!!/2018
 5/30/2018 Problem Set 8  6/4/2018

Midterm and Final Exam
Midterm available here. The password has been emailed to you. If you have any issues accessing the file, please contact Meagan. The midterm is due at 4 pm on Tuesday May 8th.

Final exam available here. The password has been emailed to you. If you have any issues accessing the file, please contact Meagan. The final is due at 4 pm on Friday June 15th.

Collaboration Table
  Homework Exams
You may consult:    
Course textbook (including answers in the back) YES YES
Other books YES NO
Solution manuals NO NO
Internet YES NO
Your notes (taken in class) YES YES
Class notes of others YES NO
Your hand copies of class notes of others YES YES
Photocopies of class notes of others YES NO
Electronic copies of class notes of others YES NO
Course handouts YES YES
Your returned homework / exams YES YES
Solutions to homework / exams (posted on webpage) YES YES
Homework / exams of previous years NO NO
Solutions to homework / exams of previous years NO NO
Emails from TAs YES NO
You may:

Discuss problems with others YES NO
Look at communal materials while writing up solutions YES NO
Look at individual written work of others NO NO
Post about problems online NO NO
For computational aids, you may use:

Calculators YES* NO
Computers YES* NO

* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work.  For example, "by Mathematica" is not an acceptable justification for deriving one equation from another.  Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them.

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