Absences
Nathan: Sept 16 - 19.
Chris: Sept 13.
Jeremiah: Two weeks starting around October 13.
Iftikhar:

# Detailed schedule for Math 241.

## Math 241: Week 1

Aug 26
Introduction (§12.1). Notes.
HW 1 (Due Fri Aug 30)

2016 and 2018: 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. It might be best to cut 5 minutes from the lecture to have more time with the syllabus. 2019: Started as soon as the bell rung, which made everything fit more easily.

Aug 27
Review of parts of Calc I and II. Worksheet. Solutions.

Final exam schedule request due to Aaron.

Aug 28
Vectors (§12.2) and the dot product (§12.3). Notes.
HW 2 (Due Tue Sept 3 because of Labor Day.)

2016: Did not get to the derivation of the dot product formula on page 6; that page is not to be covered. 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. 2018: cut position vectors and some properties but still had only 10 minutes for the dot product. 2019: same cuts as 2018 but had 20 minutes (first hour) and 15 minutes (second hour) for the dot product. Also, while we usually think of theta as the smaller of the two angles on page 5, the formula also works for the other one as the cosines are the same.

Aug 29
Parametric curves via vector arithmetic. Worksheet. Solutions.
Aug 30
Dot product applications (§12.3) and equations for planes (§12.5). Notes.
HW 3 (due Wed, Sept 4)

2016: 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. 2018: Skipped regression and completed the section on projections and work in about 15 minutes. That left enough time for a full treatment of the rest. 2019: in both hours, started the plane examples/problems (bottom of page 4), with 20 minutes left. This was OK, but the answer A2 was packed into just a couple minutes. There's a typo on page 6: r(t) = (t, -t, 1).

Notes:
Pickup syllabus if you weren’t here on Monday.
No class Monday, Sept 2 (Labor Day).
Class website: http://dunfield.info/241.

## Math 241: Week 2

Sept 2
Labor Day, no class and HW 2 is due tomorrow not today.
Sept 3
Projections, distances, and planes. Worksheet. Solutions.
Sept 4
Cross product (§12.4). Notes.
HW 4 (Due Mon, Sept 9)

2016: In both sections, got through the plane example but not the triple product or anything after that. 2018 and 2019: Got through both the plane example and the triple product, but only had 5 minutes for each.

Sept 5
Vectors and the geometry of 3-space discussion.

Assign exam rooms based on current enrollments (Nathan)

Sept 6
Functions of several variables (§14.1). Notes.
Hopf fibration video.
HW 5 (Due Wed, Sept 11)

2016: In both sections, got through everything with essentially no time to spare. 2018: I tried both ways, and it is best to skip the plane example to have more time to talk about the cool video. 2019: Both times skipped the plane example and hit the 3-variable function with 10 minutes left.

Exam 1 draft due (Iftikhar and Jeremiah)

Notes:
Class website: http://dunfield.info/241.
If you have a laptop or tablet, please bring it to section next week on Tuesday.

## Week 3

Sept 9
Level sets in 3d (§14.1); quadric surfaces (§12.6); intro to limits (§14.2). Notes.
More on limits.
HW 6 (Due Fri, Sept 13)

2016: 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. 2018 and 2019: Using the Gallery to breeze through quadrics, started in on limits at 20 past, did everything except the example on the middle of page 6.

Sept 10
If you have a laptop or tablet, please bring it to section today.
Sept 11
Limits in several variables (§14.2). Notes.
Extras: Limit pictures; The sinking of the Sleipner A.
HW 7 (Due Mon, Sept 16)

2016 and 2018: 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.

Exam 1 to printer.

Sept 12
Functions of several variables; Limits. Worksheet. Solutions.
Sept 13
Limit laws (§14.2); Continuity in several variables (§14.2); partial derivatives (§14.3). Notes.
HW 8 (Due Wed, Sept 18)
Nathan's office hour today cancelled.

2016: 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. 2018: Also got through page 6 in both sections, making a conscious effort to keep the pace up. Started partial derivatives with 10-13 minutes left depending on the section.

Notes:
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 24 from 6:45–8:00pm.
See here for complete details, including how to register for the conflict exam.

## Week 4

Sept 16
Applications of partial derivatives (§14.3 and §14.4). Notes.
Visualization: Heat equation.
HW 9 (Due Fri, Sept 20)

2016: 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. 2018: Both times I got through everything except the proof of the first theorem at the bottom of page 5 with a minute or two to spare.

Sept 17
Partial derivatives and differentiability. Worksheet. Solutions.
Nathan's office hour today cancelled.
Sept 18
Chain rule (§14.5). Notes.
HW 10 (Due Mon, Sept 23)

Note: There are two examples in the notes, on pages 4 and 5. The second one involves slightly less writing and takes a little less time to do.

2016: 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 theoretically, 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.

2018: I dropped the explicit error terms on Page 3 and used the second (shorter) example. In the first lecture, got through the page titled "Alternate viewpoint". In the second lecture, got to the same point with 5 minutes left, and just ended early.

Sept 19
Practice exam for Midterm 1. Solutions.
Nathan's office hour today cancelled.
Sept 20
More on the chain rule (§14.5), directional derivatives and the gradient (§14.6). Notes.
HW 11 (Due Fri, Sept 27)
Nathan's office hour today extended to 2:30–4:30pm

2016: 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. 2018: This was fine.

Notes:
The first midterm exam will be Tuesday, Sept 24 from 6:45–8:00pm.
See here for complete details, including how to register for the conflict exam.

## Week 5

Sept 23
More on the gradient (§14.6) and overview of optimization (§14.7–14.8). Notes.
HW 12 (Due Mon, Sept 30)

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

Sept 24
Section: Review for midterm.
Midterm the First: 6:45–8:00pm. Solutions.
Sept 25
No class.
Sept 26
Midterm discussed. Solutions.
Sept 27
Local min and max (§14.7). Notes.
HW 13 (Due Wed, Oct 2)

2018 and 2019: This was fine when I skipped any detailed discussion of why x^3 is unstable. I finished page 4 at 30 minutes in with 10 minutes left for each of pages 5 and 6.

Notes:
Different tutoring and office hours this week.

## Week 6

Sept 30
Absolute min and max (§14.7). Notes.
HW 14 (Due Fri, Oct 4)

2016: 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. 2018: Changed the final example to simplify it; was able to get through most of said example in 6 minutes and all of it in 8-9 minutes. The asides on page 5 were mostly skipped, and I have no idea what the point of the visualization is and so I deleted it for 2019. 2019: This was fine, I had about 10 minutes for the example on page 5.

Oct 1
Taylor series, the second derivative test, and changing coordinates. Worksheet. Solutions.
Oct 2
Constrained min/max (§14.8). Notes.
HW 15 (Due Mon, Oct 7)

2016, 2018, and 2019: This lecture is a good length. There is time at the end to discuss why the local max in the last example is the absolute max, so you should be prepared to explain that.

Oct 3
Constrained min/max via Lagrange multipliers. Worksheet. Solutions.
Oct 4
Introduction to space curves (§13.1–4). Notes.
Visualization: Cycloid.
HW 16 (Due Wed, Oct 9)

2016: This lecture is ok, I got though everything though I wished it was a bit shorter. 2018 and 2019: This was fine. Finished the examples and cycloid discussion at 25 minutes in.

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 7
More on arc length (§13.3) and integrating functions on curves (§16.2, pages 1063–1065). Notes.
HW 17 (Due Fri, Oct 11)

2016: 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 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. 2018 and 2019: This was fine. I needed about 15-16 minutes to do everything after "Understanding these integrals"; once I did it in 13 minutes which was kinda tight.

Oct 8
Lagrange multipliers problem discussion.

Exam 2 draft due (Chris and Nathan)

Midsemester feedback form administered in section.

Oct 9
Vector fields (§16.1) and integrating them along curves (§16.2). Notes.
Example vector field: Live wind map
HW 18 (Due Mon, Oct 14)

2016: 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. 2018: This lecture is fine. I started page 4 with 20 minutes left, and the last example with 5 minutes left, which is slightly too little. 2019: Same as 2018 but had 8 minutes left for the last example which is better.

Oct 10
Curves and integration. Worksheet. Solutions.

Should learn date of final now.

Oct 11
More on integrating vector fields along curves; the Fundamental Theorem of Line Integrals (§16.2 and §16.3). Notes.
HW 19 (Due Wed, Oct 16)

2016: 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" part of the calculation on page 5. 2018: In the first hour, I did everything and in the example on page 5 added a discussion of why we should expect the answer to be positive geometrically based on the first half of the lecture. In the second hour, I had to drop the "by hand" part of the calculation on page 5. In both hours, I started the discussion of the Fund Thm of Line Integrals with 20 minutes left.

Notes:
The final exam for Math 241 will be Friday, December 13 at 1:30pm.

## Week 8

Oct 14
Conservative vector fields I (§16.3). Notes.
HW 20 (Due Fri, Oct 18)

2016: 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. 2018: In both hours, I did everything except the last example on page 6. 2019: Went much slower this time. In the first hour had to rush to even state Theorems A and B before the bell rang. In the second hour, I stated them and applied Theorem B to (y, x) and (-y, x) but didn't even motivate Theorem B.

Oct 15
Integrating vector fields. Worksheet. Solutions.

Exam 2 to printer.

Oct 16
Conservative vector fields II (§16.3). Notes.
HW 21 (Due Mon, Oct 21)

2016: As written, this lecture is a little short. In the first hour, even after padding out the discussion 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. 2018: Similar.

Oct 17
Discussion of line integral problems.
Oct 18
Intro to multiple integrals (§15.1). Notes.
HW 22 (Due Fri, Oct 25)
Last day to drop the course (*)

2018: This is fine.

Notes:
The second midterm exam will be Tuesday, Oct 22 from 6:45–8:00pm.
See here for complete details.

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

## Week 9

Oct 21
Integrating over more complicated regions (§15.2 and §15.3). Notes.
HW 23 (Due Mon, Oct 28)

2016, 2018, and 2019: Got through notes as written. In 2019, got to polar coordinates with 20 minutes left.

Oct 22
Section: Review for midterm.
Midterm the Second: 6:45–8:00pm. Solutions.
Oct 23
No class.
Oct 24
Multivariable integrals. Worksheet. Solutions.
Oct 25
Polar coordinates (§15.3) and applications (§15.4). Notes.
HW 24 (Due Wed, Oct 30)

2016: I got through everything, 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. 2018 and 2019: Got to the section "Applications of Integration" with 20 minutes left both times, which is not quite enough, resulting in some rushing at the very end.

Notes:
Different tutoring and office hours this week.

## Week 10

Oct 28
Triple integrals (§15.6). Notes.
HW 25 (Due Fri, Nov 1)

2016: 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. 2018: Similar, and got that same question both times. 2019: Went slower this time, hit the cylindrical coordinates with 10 minutes left in the first hour and 15 in the second. 10 minutes is not enough but 15 is OK.

Oct 29
Transformations of the plane. Worksheet. Solutions.
Oct 30
Integrating in cylindrical and spherical coordinates (§15.7 and §15.8). Notes.
HW 26 (Due Mon, Nov 4)

2016: 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. 2018: I skipped the calculation on page 3 and had at least 5 minutes to spare at the end, maybe more. I improvised a brief discussion of what slices look like in spherical coordinates, which was a good addition. 2019: Skipped the calculation on page 3 and used the whole time in both sections, finishing the cylindrical example at 30 and 27 minutes in, respectively.

Oct 31
Discussion of multivariable integral problems.
Nov 1
Changing coordinates I (§15.9). Notes.
HW 27 (Due Wed, Nov 6)

2016: 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. 2018 and 2019: In both sections, I skipped talking about linear approximation and instead talked about the average.

## Week 11

Nov 4
Changing coordinates II (§15.9). Notes.
HW 28 (Due Fri, Nov 8)

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

2016: 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. 2018 and 2019: Skipping evaluating the integrals, I got through the end of page 6.

2016: 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 5
Integrating by changing coordinates. Worksheet. Solutions.

Exam 3 draft due (Chris and Jeremiah)

Nov 6
Surfaces in R3 (§16.6). Notes.
HW 29 (Due Mon, Nov 11)

2016: This lecture is fine, even with the 5 minutes at the beginning for the general 3D change of coordinate formula. 2018 and 2019: This lecture is fine.

Nov 7
Surface Parameterpolooza. Worksheet. Solutions.
Nov 8
Area and integration on surfaces (§16.6 and §16.7). Notes.
Visualization: Cones.
HW 30 (Due Wed, Nov 13)

2016: 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. 2018: Even condensing the calculations, I had a hard time creating more than 6-7 minutes for the visualization.

## Week 12

Nov 11
Green’s Theorem (§16.4). Notes.
HW 31 (Due Fri, Nov 15)

2016, 2018, and 2019: This lecture was fine, maybe even a little short.

Nov 12
Parametrizations and integrals. Worksheet. Solutions.

Exam 3 to printer.

Nov 13
Green’s Theorem and conservative vector fields in 2D (§16.4), but mostly flux in 2D (§16.5). Notes.
Visualization: Flux and flow.
HW 32 (Due Mon, Nov 18)

2016: 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. 2018: In both sections, I skipped the non-simple curves case of Green's theorem and still could only do the bit of page 6 where I derived the formula for the normal from the one for the velocity vector.

Nov 14
Green's Theorem. Worksheet. Solutions.
Nov 15
The Divergence Theorem in 2D (§16.5-16.9). Notes.
HW 33 (Due Fri, Nov 22)

2016: This lecture was an OK length given that I condensed the bit on page 3 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. 2018: Revised page 3 to use the language of transformations. In both sections, got to the separator on page 5, but nothing about how this is equivalent to Green's Theorem. 2019: Got through page 5 both times, including the part past the seperator, though I had only 5 minutes to sketch that.

Notes:
The third midterm exam will be Tuesday, Nov 19 from 6:45–8:00pm.
See here for complete details.

## Week 13

Nov 18
Surface integrals of vector fields and the divergence theorem in 3D (§16.7 and §16.9). Notes.
HW 34 (Due Mon, Dec 2)

2016: 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. 2018: I just did the first four pages, which quite easily filled 50 minutes; if one has a five minutes at the end, one could redo the solid cone example with another vector field, e.g. (x, 0, 0) or the easier (0, 0, z) so that the divergence theorem tells us we are computing the volume.

Nov 19
Section: Review for Midterm 3.
Midterm the Third: 6:45–8:00pm. Solutions.
Nov 20
Parameterizing the real world: surfaces in computer-aided design. Notes. Lecture video.
Visualization: Bezier curves and surfaces.
Online visualization: Bezier curves.
Recent research: CARGO project, additional paper
HW: None

Draft of part 1 of final, covering roughly midterms 1 and 2 due. (Mostly Nathan and Iftikhar)

Nov 21
Midterm discussed. Solutions.
Nov 22
No class, but HW 33 is due.
Notes:
Different tutoring and office hours this week.

## Week 14

Dec 2
Stokes Theorem (§16.8), including the definition of the curl (§16.5). Notes.
HW 35 (Due Fri, Dec 6).

Draft of part 2 of final, covering roughly midterm 3 and after. (Mostly Chris and Jeremiah)

This lecture is much longer than it looks. I made one change: moved computing the curl of the vector field in the example, namely F = (-y, x, yz), forward to page 2 since it's more generic than than F = (y, 0, 0), which I also did. Only putting the top half of the first page on the board to start, I hit the top of page 4 with only 10 minutes left both times, which is tight even skipping the bit at the end about the lower hemisphere. 2019: Same as last time.

Dec 3
Surface integrals of vector fields. Worksheet. Solutions.

TAs should do ICES today.

Dec 4
More on Stokes Theorem (§16.8), including understanding the curl (§16.5). Review of conservative vector fields. Notes.
HW 36 (Due Mon, Dec 9)

2016, 2018 and 2019: This lecture is a good length.

Dec 5
Stokes’ Theorem. Worksheet. Solutions.
Dec 6
Conservative vector fields in R3 (§16.8); Topology 101. Notes.
HW 37 (Due Wed, Dec 11)

2016 and 2018: This lecture was a fine length. 2019: In the first hour, I almost ran out of material; in the second hour, I didn't quite get through it all. Both results were fine.

## Week 15

Dec 9
Electrostatics and Gauss’s Law. (§16.9) Notes.

2016: 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. 2018: It took me 45 minutes, which left less time for ICES than I'd like. Suggest just skipping the calculation of div E in future years. 2019: Skipping calculating div F, it still took 42-43 minutes but I did talk about the second half of the last page.

Instructors should do ICES today.

Final exam to printers.

Dec 10
More on Stokes’ Theorem. Worksheet. Solutions.
Dec 11
Maxwell’s equations. Notes.

2016: 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.

Dec 12