Detailed schedule for Math 241.

Math 241: Week 1

Aug 22
Introduction (§12.1). Notes.
HW 1 (due Fri Aug 26)

Without any digressions, took 35 minute to cover this, leaving 10 minutes for going over the syllabus and 5 for starting a little late. When covering distance in n-dimensions, it would be better to first compute the distance for the origin O to P and also to label the points as O and P in the 2D and 3D examples.

Aug 23
Review of parts of Calc I and II. Worksheet. Solutions.

Final exam schedule request due to Aaron.

Aug 24
Vectors (§12.2) and the dot product (§12.3). Notes.
HW 2 (Due Mon, Aug 29)

Did not get to the derivation of the dot product formula on page 6. Also hit the dot product 40 minutes in. Next time cut something to so I will have 12.5-15 minutes for the dot product. For example, position vectors and some of the properties of vector operations.

Aug 25
Parametric curves via vector arithmetic. Worksheet. Solutions.
Aug 26
Dot product applications (§12.3) and equations for planes (§12.5). Notes.
HW 3 (due Wed, Aug 31)

This lecture is way too long. The first time through, I had to skip parameterizing the line L entirely! The second time, I skipped the discussion of regression and got through more, but it was still tight.

Pickup syllabus if you weren’t here on Monday.
Class website:

Math 241: Week 2

Aug 29
Cross product (§12.4). Notes.
HW 4 (Due Fri, Sept 2)

In both sections, got through the plane example but not the triple product or anything after that.

Aug 30
Projections, distances, and planes. Worksheet. Solutions.
Aug 31
Functions of several variables (§14.1). Notes.
Hopf fibration video.
HW 5 (Due Tue, Sept 6)

In both sections, got through everything with essentially no time to spare.

Sept 1
Vectors and the geometry of 3-space discussion. Quiz on HW 1–3.

Assign exam rooms based on current enrollments (Nathan)

Sept 2
Level sets in 3d (§14.1); quadric surfaces (§12.6); intro to limits (§14.2). Notes.
Interactive guide to quadric surfaces.
More on limits.
HW 6 (Due Wed, Sept 7)

Got through the proof that h^2 has limit 0 at 0 in both sections, but not h^2 + 2h. In the 8am, had no time to spare, in the 9am had about 3 minutes. In both cases, started limits between 20 and 25 minutes in (closer to 20, I think). The discussion of quadric surfaces types was done quickly by showing the Interactive Gallery and referring them to the upcoming discussion section on this topic.

Exam 1 draft due (Nathan and Sue)

Tutoring room moved to 343 Altgeld Hall.
Class website:
No class Monday, Sept 5 (Labor Day); my office hour on Tuesday, Sept 6, is cancelled.
If you have a laptop or tablet, please bring it to section next week on Tuesday.

Week 3

Sept 5
Labor Day, no class and HW 5 is due tomorrow not today.
Sept 6
Visualizing quadric surfaces. Solutions.
If you have a laptop or tablet, please bring it to section today.
Sept 7
Limits in several variables (§14.2). Notes.
Extras: Limit pictures; The sinking of the Sleipner A.
HW 7 (Due Mon, Sept 12)

In both sections, got up to the top of page 5, just before "rules for limits" with time enough to do the limit pictures Mathematica notebook and tell the sad story of the Sleipner A.

The description of HW 7 is "This HW covers questions on limits (14.2), but not continuity" but the very first problem (and only that problem) involves continuity! Move this problem to the next assignment in future years.

Exam 1 to printer.

Sept 8
Functions of several variables; Limits. Worksheet. Solutions.
Sept 9
Limit laws (§14.2); Continuity in several variables (§14.2); partial derivatives (§14.3). Notes.
HW 8 (Due Wed, Sept 14)

I was under the weather a bit, and in that context the lecture seemed a bit long. I did get through page 6 in both sections, but I really had to keep an eye on the clock and push it to make this happen.

Having difficulty copying everything down in class? You can always fill stuff in from the lecture notes above.
The first midterm exam will be Tuesday, Sept 20 from 7:45–9:00pm.
See here for complete details, including how to register for the conflict exam.

Week 4

Sept 12
Applications of partial derivatives (§14.3 and §14.4). Notes.
Visualization: Heat equation.
HW 9 (Due Fri, Sept 16)

The first lecture went poorly because of computer glitches. In the second lecture, I got through everything except the proof of the first theorem at the bottom of page 5. I had to keep the pace up for that to happen, however, so as usual it would be nicer if the lecture was shorter.

Sept 13
Partial derivatives and differentiability. Worksheet. Solutions.
Sept 14
Chain rule (§14.5). Notes.
HW 10 (Due Mon, Sept 19)

In the first lecture, got through page 5 only by working at a demandingly quick pace. In the second lecture, ran out of time halfway through page 6. Overall, there's a lot to write out in this lecture, and overall the lecture is a bit long. I suspect the whole derivation of the Chain Rule is too messy for them to get much out of; on the other hand, it is a nice of example of using linear approximation theorically, which will occurs frequently in these lectures. Perhaps one should derive the single-variable chain rule and then just talk about what happens for two variables in a more heuristic fashion.

HW 10 should make clear in the intro that some of the problems, namely those involving the Chain Rule for partial derivatives, depend on the very beginning of the next lecture.

Sept 15
Practice exam for Midterm 1. Solutions.
Sept 16
More on the chain rule (§14.5), directional derivatives and the gradient (§14.6). Notes.
Visualization: Gradient and level sets.
HW 11 (Due Fri, Sept 23)

This lecture is a nice length; even though I expanded the first part slightly to deal with leftover material from the previous lecture, I was able to get through it all without working too hard.

The first midterm exam will be Tuesday, Sept 20 from 7:45–9:00pm.
See here for complete details, including how to register for the conflict exam.

Week 5

Sept 19
More on the gradient (§14.6) and overview of optimization (§14.7–14.8). Notes.
HW 12 (Due Mon, Sept 26)

This lecture is a nice length and I was able to get through it all without working too hard.

Sept 20
Section: Review for midterm.
Midterm the First: 7:45–9:00pm. Solutions.
Sept 21
No class.
Sept 22
Midterm returned and discussed. Solutions.
Sept 23
Local min and max (§14.7). Notes.
HW 13 (Due Wed, Sept 28)
Different tutoring and office hours this week.

Week 6

Sept 26
Absolute min and max (§14.7). Notes.
Visualization: Min/max example.
HW 14 (Due Fri, Sept 30)

In the first hour, I got through the top of page 6 fine and then tried to cram the last example into 3 minutes. In the second hour, I skipped the top of page 5 and so had more like 5-6 minutes for the last example, which was still too little to cover it completely. Also, in both sections, I forgot about the Mathematica visualization.

Sept 27
Taylor series, the second derivative test, and changing coordinates. Worksheet. Solutions.
Sept 28
Constrained min/max (§14.8). Notes.
HW 15 (Due Mon, Oct 3)

This lecture is a good length. In the first section, I even got through a reasonably complete discussion of why the local max is the absolute max. In the second section, I didn't have time for the absolute max discussion, but I didn't have everything (including the contour plot) up beforehand as I was answering questions during the break.

HW 15 says: "This HW covers Lagrange multipliers, but not multiple constraints (14.8)" but in fact the final problem has two constraints. This should be resolved somehow.

Sept 29
Constrained min/max via Lagrange multipliers. Worksheet. Solutions.
Sept 30
Introduction to space curves (§13.1–4). Notes.
Visualization: Cycloid.
HW 16 (Due Wed, Oct 5)

This lecture is ok, I got though everything though I wished it was a bit shorter.

In detail, this is §13.1, the first subsection of §13.2 (through page 849), the first subsection of §13.3 on arc length (through page 855.5), and §13.4 through page 863.

Week 7

Oct 3
More on arc length (§13.3) and integrating functions on curves (§16.2, pages 1063–1065). Notes.
HW 17 (Due Fri, Oct 7)

This lecture is actually slightly short for once. I padded it out with a discussion of what the integral of x over the 1/4 circle C is means in terms of an area, and also said that "ds" is called the "arc-length element", which is at the beginning of the next notes. Should have included that the integral of 1 is the length.

Oct 4
Lagrange multipliers problem discussion. Quiz on HW 11–15.

Exam 2 draft due (Sue and James)

Oct 5
Vector fields (§16.1) and integrating them along curves (§16.2). Notes.
Example vector field: Live wind map
HW 18 (Due Mon, Oct 10)

This lecture is a good length. It was 3-4 minutes short for the first hour and exactly right for the second. In the first lecture, I padded it out by extending the last example with a parameterization of the same curve going the opposite direction.

Oct 6
Curves and integration. Worksheet. Solutions.

Should learn date of final now. Decide on format, that is, all multiple-choice or not, and announce to all involved.

Oct 7
More on integrating vector fields along curves; the Fundamental Theorem of Line Integrals (§16.2 and §16.3). Notes.
HW 19 (Due Wed, Oct 12)

In the first hour, I did everything except page 6 on independence of path. In the second hour, I did page 6 in response to a question, but didn't do the "by hand" calculation on page 5.

The final exam for Math 241 will be Tuesday, December 13 from 1:30-4:30pm.

Week 8

Oct 10
Conservative vector fields I (§16.3). Notes.
HW 20 (Due Fri, Oct 14)

In the first hour, I did everything except the last example on page 6. In the second hour, I did everything with 3 minutes to spare. One minor thing: on page 2, the example is not quite the same as in Lecture 18: they differ by a factor of 2.

Oct 11
Integrating vector fields. Worksheet. Solutions.

Exam 2 to printer.

Oct 12
Conservative vector fields II (§16.3). Notes.
HW 21 (Due Mon, Oct 17)

As written, this lecture is a little short. In the first hour, even after padding out the dicussion of why averages over shorter and shorter paths converged to the value of the function at the fixed endpoint, I actually ended class 5 minutes early. In the second hour, I did take the whole period, but I was definitely moving more slowly than usual. Also, this is among the most theoretical lectures of the whole semester.

Oct 13
Discussion of line integral problems. Quiz on HW 16–19.
Oct 14
Intro to multiple integrals (§15.1 and §15.2). Notes.
HW 22 (Due Fri, Oct 21)
Last day to drop the course (*)
The second midterm exam will be Tuesday, Oct 18 from 7:45–9:00pm.
See here for complete details.

(*) Unsure whether to drop Math 241? Try taking the practice test for the next midterm under timed conditions to see how you’re doing.

Week 9

Oct 17
Integrating over more complicated regions (§15.3 and §15.4). Notes.
HW 23 (Due Mon, Oct 24)

Got through notes as written.

Oct 18
Section: Review for midterm.
Midterm the Second: 7:45–9:00pm. Solutions.
Oct 19
No class.
Oct 20
Midterm returned. Discussion topic: Multivariable integrals. Worksheet. Solutions.
Oct 21
Polar coordinates (§15.4) and applications (§15.5). Notes.
HW 24 (Due Wed, Oct 26)

I got through everything both times, though I had to rush the "Note" on page 6, which describes how, when the density is constant, that the center of mass is just the averages of coordinates over the region.

Different tutoring and office hours this week.

Week 10

Oct 24
Triple integrals (§15.7). Notes.
HW 25 (Due Fri, Oct 28)

This lecture is fine, I had a few minutes to spare in both sections, and that's with an extended answer to a question that came up in both sections, namely is why phi only goes from 0 to pi.

Oct 25
Transformations of the plane. Worksheet. Solutions.
Oct 26
Integrating in cylindrical and spherical coordinates (§15.8 and §15.9). Notes.
HW 26 (Due Mon, Oct 31)

This lecture is also fine, namely a few minutes short. I was feeling a little under the weather, so I skipped the straightforward calculation of the integral on page 3.

Oct 27
Discussion of multivariable integral problems. Quiz on HW 22–24.
Oct 28
Changing coordinates I (§15.10). Notes.
HW 27 (Due Wed, Nov 2)

This lecture is also fine, especially as it's not necessary to discuss the linear approximation bit at the end in any detail. In fact, I skipped it in the second hour, in favor of a brief discussion of how the average of x - y over the triangle R is thus 1/3 and why that makes geometric sense.

Week 11

Oct 31
Changing coordinates II (§15.10). Notes.
HW 28 (Due Fri, Nov 4)

Note: The 7th and last page of the notes is for reference only.

In both sections, I was unable to get through the general form of change of coordinates for triple integrals, even skipping the detailed evaluation of the integrals in the extended examples.

While personally I find the discussion of linear approximation for functions from R2 to R2 very satisfying, my guess is that it goes over their heads given how little exposure they have had with linear transformations.

Nov 1
Integrating by changing coordinates. Worksheet. Solutions.

Exam 3 draft due (Nathan and Kim)

Nov 2
Surfaces in R3 (§16.6). Notes.
HW 29 (Due Mon, Nov 7)

This lecture is fine, even with the 5 minutes at the beginning for the general 3D change of coordinate formula.

Nov 3
Surface Parameterpolooza. Worksheet. Solutions.
Nov 4
Area and integration on surfaces (§16.6 and §16.7). Notes.
Visualization: Cones.
HW 30 (Due Wed, Nov 9)

For the "last time" part, Instead of the example of the torus in the notes, put up the unit sphere, including the formulas for r_theta and r_phi. By condensing some of the calculations, I was able to get through these notes in 40 minutes, leaving 10 minutes to discuss the visualization.

Week 12

Nov 7
Green’s Theorem (§16.4). Notes.
HW 31 (Due Fri, Nov 11)

This lecture was fine, maybe even a little short.

Nov 8
Parametrizations and integrals. Worksheet. Solutions.

Exam 3 to printer.

Nov 9
Green’s Theorem and conservative vector fields in 2D (§16.4), but mostly flux in 2D (§16.5). Notes.
Visualization: Flux and flow.
HW 32Y (Due Mon, Nov 14)

This lecture is too long as written, even assuming all the "previously" bits are up before the starting bell rings. In one lecture, I skipped page 6 and in the other the "Case 2" of non-simple curves the proof of Green's Theorem. Certainly the latter should be skipped, but even then page 6 will be rushed.

Curl and Divergence (§16.5).Notes

Nov 10
Green's Theorem. Worksheet. Solutions.
Nov 11
The Divergence Theorem in 2D (§16.5-16.9). Notes.
HW 33Y (Due Fri, Nov 18)

This lecture was an OK length given that I condensed the bit on page 4 labeled "Consider skipping" and didn't do most of page 6. Overall, I feel like the basic structure of the lecture is fine, but some of the details could be explained better. Possibly one should use more the language of derivatives as opposed to that of linear approximation.

2D flux and divergence: vector form of Greens Theorem (§16.5). Notes

The third midterm exam will be Tuesday, Nov 15 from 7:45–9:00pm.
See here for complete details.

Week 13

Nov 14
Surface integrals of vector fields and the divergence theorem in 3D (§16.7 and §16.9). Notes.
HW 34Y (Due Wed, Nov 30)

Even going quickly, I could only devote 10 minutes to the heat equation stuff at the end, which isn't enough time to get through it all. If you slow down a bit, the first four pages of the notes can fill basically the whole 50 minutes.

Nov 15
Section: Review for Midterm 3.
Midterm the Third: 7:45–9:00pm. Solutions.
Nov 16
Parameterizing the real world: surfaces in computer-aided design. Notes.
Lecture video.
Visualization: Bezier curves and surfaces.
Recent research: CARGO project, additional paper

Stokes theorem part 1 (§16.8). Notes

Nov 17
Midterm returned and discussed. Solutions.
Nov 18
No class, but HW 33Y due.

Draft of part 1 of final, covering roughly midterms 1 and 2 due. (Mostly Sue and James?)

Different tutoring and office hours this week.

Week 14

Nov 28
Stokes Theorem (§16.8), including the definition of the curl (§16.5). Notes.
HW 35Y (Due Fri, Dec 2).

Stokes theorem part 2 (§16.8).Notes

Draft of part 2 of final, covering roughly midterm 3 and after. (Mostly Nathan and Kim?)

Nov 29
Surface integrals of vector fields. Worksheet. Solutions.
Nov 30
More on Stokes Theorem (§16.8), including understanding the curl (§16.5). Review of conservative vector fields. Notes.
HW 36Y (Due Mon, Dec 5)

This lecture is a good length.

Divergence Theorem (§16.9). Notes

Dec 1
Stokes’ Theorem. Worksheet. Solutions.

This worksheet was new, and in most sections they only got through the (very long) first problem. Which is a worthwhile problem, but there's some other great stuff in there as well, so might want to revise. Or put some of it into the HW?

Final exam to printers

Dec 2
Conservative vector fields in R3 (§16.8); Topology 101. Notes.
HW 37Y (Due Wed, Dec 7)

This lecture was a fine length.

Gauss’s Law (§16.9).Notes

Week 15

Dec 5
Electrostatics and Gauss’s Law. (§16.9) Notes.

This lecture is intensionally short so that there is time to do the ICES survey. It took me 40 minutes when I skipped the second half of the last page.

Conservative vectors fields in R3, differential forms, and the general Stokes theorem.Notes

Dec 6
Review for final.
Dec 7
Maxwell’s equations. Notes.

Skipping the second half of page 2 and simplifying Ampere's law to the case when the current is 0 made the notes as written take 35 minutes, leaving 15 minutes for a stirring summary the integral theorems and a hint at the general form of Stokes' Theorem.

Practical review for final

Dec 8
Reading day.
Dec 9
Finals begin.
Extra office hours and tutoring later in the week, see here for complete details.

Week 16

Dec 12
Dec 13
Final the Ultimate.
For office hours and tutoring this week see here for complete details.

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