Reading Assignments / Lecture Schedule / News and Updates
- Fri 12/8 FINAL EXAM 2pm-5pm in CIT 227
Final Exam Topics and review problems. Updated 12/6: some formulas added/fixed on the formulas sheet.
- Wed 12/8 1pm-3pm in Wilson 304: Review session/extended office hours. Come with questions/example problems you'd like to see solved. I'd prefer to have these review sessions driven by you, rather than me.
- Mon 12/6 10am-12pm in Wilson 205: Review session/extended office hours. Come with questions/example problems you'd like to see solved. I'd prefer to have these review sessions driven by you, rather than me.
- Wed 12/1 and Fri 12/3: Differential Forms and Vector Calculus in higher dimensions. Ch 8.6
This material will not be covered on the final exam.
- Mon 11/29: Electromagnetism and Maxwell's Equations. I'll show how to derive Maxwell's equations from a few experimentally verified laws
using our vector calculus tools, and some of their interesting implications, both mathematical and physical.
This material will not be covered on the final exam.
- No class on Wed 11/24 or Fri 11/26.
- Mon 11/22: Summary of the theorems of vector calculus, example calculations. Ch 8.
- Fri 11/19: The Divergence Theorem (Gass' Theorem), Gauss' Law. Ch 8.4
- Wed 11/17: More on Stokes' Theorem, curl and conservative fields. Ch 8.3
Here is the proof of the 4 equivalent characterizations of conservative vector fields. I screwed this up a bit in the 10am lecture.
- Mon 11/15: More examples w/ Green's Theorem, Stokes' Theorem. Ch 8.2
- Fri 11/12: Proof of Green's Theorem, examples. Ch 8.1
- Wed 11/10: Divergence and curl continued, flux in 2D, staring on Green's Theorem. Ch 8.1
- Mon 11/8: Return and go over Exam II. Introducing the `del' operator, curl and divergence. Ch 4.4
- Fri 11/5: Exam II in class.
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.
- Wed 11/3: Review for Exam II. Ch 5, 6, 7
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.
- Mon 11/1: Surface integrals of vector fields, orientation. Ch 7.6
- No class Friday 10/29
No office hours on Thursday 10/28 either. I will be out of town.
- Wed 10/27: Parametrized surfaces, surface integrals of scalar functions. Ch 7.3, 7.4, 7.5
- Mon 10/25: Line integrals continued, parametrization independence, line integrals of gradient fields.
We might begin surface area and surface integrals. Ch 7.2, maybe parts of 7.3/4
- Fri 10/22: Arc length and path integrals continued, vector fields and line integrals. Ch 4.3, 7.1, 7.2
- Wed 10/20: Improper integrals. Introducing arc length and path integrals. Ch 6.4, 4.1, 4.2, 7.1
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.
- Mon 10/18: Applications of multiple integrals, improper integrals. Ch 6.3, 6.4
- Fri 10/15: Change of variables proof sketch. Ch 6.2
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.
- Wed 10/13: Change of variables in double and triple integrals. Ch 6.1, 6.2
- Fri 10/8: Changing the order of integration, triple integrals. Ch 5.4, 5.5.
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.
- Wed 10/6: Fubini's Theorem, integration over elementary regions, changing the order of integration. Ch 5.3, 5.4.
- Mon 10/4: Return and go over exam. Double integral over rectangles, properties of the integral, maybe Fubini's Theorem. Ch 5.1, 5.2.
- Fri 10/1: Exam I in class.
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.
- Wed 9/29: Review for Exam I.
Here is a list of topics from which Exam I questions will be taken.
- Mon 9/27: Lagrange multipliers with several constraints, second derivative test for Lagrange multipliers, implicit and inverse function
theorems. Ch 3.4, 3.5.
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.
- Fri 9/24: Lagrange multipliers. Ch 3.4.
- Wed 9/22: Local and global extrema continued, second derivative test in higher dimensions, introducing Lagrange multipliers. Ch 3.3, 3.4 (pp. 225-232).
- Mon 9/20: Local and global extrema, second derivative test for max/min. Ch 3.3, 1.3.
In discussing the second derivative test, we'll use determinants of 2x2 matrices, which is covered in Ch 1.3.
- Fri 9/17: Taylor's Theorem for functions of several variables, first derivative test for local extrema. Ch 3.2, Ch 3.3 (pp. 207-210).
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.
- Wed 9/15: Iterated partial derivatives, Taylor's Theorem. Ch 3.1, 3.2.
- Mon 9/13: Gradients and curves revisited, tangent planes and lines. Reading: anything you haven't yet read from Ch 2.
- Fri 9/10: The total derivative, properties of derivatives. Same reading as last time.
I typed up the proof of the chain rule, which you can download here.
- Wed 9/8: Matrices and matrix multiplication, the total derivative, properties of derivatives. Ch 1.5 (pp. 87-83), Ch 2.3, 2.5.
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.
- Fri 9/3: Limits continued, continuity, partial derivatives, gradient, curves and velocity. Ch 2.3 (pp. 131-140 will be covered
next Wednesday), 2.4, 2.6, 1.1, 1.2.
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.
- Wed 9/1: Functions of several variables, limits. Ch 2.1-2.2 (pp. 107-115 optional).
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.