Final Exam Topics and review problems. Updated 12/6: some formulas added/fixed on the formulas sheet.

This material will not be covered on the final exam.

This material will not be covered on the final exam.

Here is the proof of the 4 equivalent characterizations of conservative vector fields. I screwed this up a bit in the 10am lecture.

Update 11/7: Grades have been posted to mycourses. Though you do not officially recieve letter grades on this exam, approximate letter grade ranges would be something like A: 45-60, B: 35-44, C: 25-34, D: 20-24, F:0-19.

Here is a list of topics from which Exam II questions will be taken. On page 3 is a list of formulas which you will be given on the exam.

No office hours on Thursday 10/28 either. I will be out of town.

We'll start to cover some of the material from Chapter 4, picking it up as we need it for the material in Chapter 7.

My apologies for this lecture. I got pretty jammed up in both morning and afternoon lectures, and the proof sketch I tried to give contained some errors.

Since Monday is a holiday, there will be no Monday office hours. I will have extra office hours both from 11-12 and 1-2 on Wednesday.

Update 10/2: Grades have been posted to mycourses. Total is out of 88 (Problem #4 was regraded out of 8 instead of 20). Though you do not officially recieve letter grades on this exam, approximate letter grade ranges would be something like A: 70-88, B: 60-69, C: 50-59, D: 35-49, F: 0-34.

Here is a list of topics from which Exam I questions will be taken.

Just a friendly reminder that Exam 1 will take place in class this Friday, Oct. 1. It will cover everything we've done so far, including Lagrange multipliers, but not including the implicit or inverse function theorems. This corresponds to Chapters 2 and 3 of the book, except 3.5 (and also not including the stuff about open and closed sets in Chapter 2). You will also need whatever background from Chapter 1 we've been using.

Note that the second derivative test (either the usual one, or for Lagrange multipliers) can only confirm that critical points are maxima or minima. It does not rule out the possibility of maxima or minima if it fails. I may have stated this incorrectly in lecture.

My official office hours are now Mondays and Thursdays from 11-12, Wednesdays from 1-2. I will no longer have regular office hours on Fridays.

In discussing the second derivative test, we'll use determinants of 2x2 matrices, which is covered in Ch 1.3.

If you haven't yet, read Ch 1.5. It contains a pretty good account of matrices and linear maps from R^n to R^m.

I typed up the proof of the chain rule, which you can download here.

Mistake: The definition I gave last time of ``differentiable'' should really have been called ``continuously differentiable'' (also known as class C^1). I'll give you the definition of differentiability on Wednesday. By Theorem 9 (p. 137), a function which is continuously differentiable is differentiable, but the converse does not generally hold.

Since Monday is a holiday and I will not be on campus, I will have office hours both from 11-12 and 1-2 on Wednesday 9/8.

Since Wednesday was rather abstract, I thought we'd come down to earth for a bit and cover gradients and velocity (which are the total derivatives of scalar functions and curves, respectively) before we talk about the total derivative for the most general case of a vector valued function of several variables, which we'll cover next Wednesday. This is out of order with respect to the textbook, which does the general case first. I recommend you read all of section 2.3, but don't worry too much about understanding everything in pp. 131-140 yet.

We'll also need the notion of the dot product of vectors, discussed in 1.2. I'll give the definition briefly in class, but you should study the text if you've never seen this before.

For those of you who were a bit uncomfortable with the function and set notation on Wednesday, have a look at p. xxvi in the book, where most of the notation is explained.

Note that we're using the epsilon-delta characterization as the definition of the limit, whereas M-T use an open set definition, and then state a theorem that the epsilon-delta characterization is equivalent.

Note also that in 10am lecture, I swapped epsilon and delta. The definition is still correct, but it is Very Bad Form to go against the usual convention, which is as stated in M-T on page 121, Theorem 6. I'll correct this next time, and you may want to reverse these in your notes to avoid more confusion in the future.