James: Friday Sept 7; October 29 - 31 (Monday - Wednesday);
Patrick: The week of Sept 17 - 21;

Detailed schedule for Math 241.

Math 241: Week 1

Aug 27
Introduction (§12.1). Notes.
HW 1 (Due Fri Aug 31)

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.

Aug 28
Review of parts of Calc I and II. Worksheet. Solutions.

Final exam schedule request due to Aaron.

Aug 29
Vectors (§12.2) and the dot product (§12.3). Notes.
HW 2 (Due Tue Sept 4 because of Labor Day.)

2016: 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. 2018: cut position vectors and some properties but still had on 10 minutes for the dot product.

Aug 30
Parametric curves via vector arithmetic. Worksheet. Solutions.
Aug 31
Dot product applications (§12.3) and equations for planes (§12.5). Notes.
HW 3 (due Wed, Sept 5)

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.

Pickup syllabus if you weren’t here on Monday.
No class Monday, Sept 3 (Labor Day).
Class website: http://dunfield.info/241.
HW: http://www.webassign.net/uiuc/login.html

Math 241: Week 2

Sept 3
Labor Day, no class and HW 2 is due tomorrow not today.
Sept 4
Projections, distances, and planes. Worksheet. Solutions.
Sept 5
Cross product (§12.4). Notes.
HW 4 (Due Mon, Sept 10)

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

Sept 6
Vectors and the geometry of 3-space discussion. Quiz on HW 1–3.

Assign exam rooms based on current enrollments (Nathan)

Sept 7
Functions of several variables (§14.1). Notes.
Hopf fibration video.
HW 5 (Due Wed, Sept 12)

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.

Exam 1 draft due (Nathan and James)

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 10
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 Fri, Sept 14)

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: Started in on limits at 20 past, did everything except the example on the middle of page 6.

Sept 11
Visualizing quadric surfaces. Solutions.
If you have a laptop or tablet, please bring it to section today.
Sept 12
Limits in several variables (§14.2). Notes.
Extras: Limit pictures; The sinking of the Sleipner A.
HW 7 (Due Mon, Sept 17)

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 13
Functions of several variables; Limits. Worksheet. Solutions.
Sept 14
Limit laws (§14.2); Continuity in several variables (§14.2); partial derivatives (§14.3). Notes.
HW 8 (Due Wed, Sept 19)

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 consconscious effort to keep the pace up. Started partial derivatives with 10-13 minutes left depending on the section.

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

Week 4

Sept 17
Applications of partial derivatives (§14.3 and §14.4). Notes.
Visualization: Heat equation.
HW 9 (Due Fri, Sept 21)

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 18
Partial derivatives and differentiability. Worksheet. Solutions.
Sept 19
Chain rule (§14.5). Notes.
HW 10 (Due Mon, Sept 24)

Note: There are two examples in the notes, on pages 4 and 5. The second one involves slightly less writting 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 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.

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 20
Practice exam for Midterm 1. Solutions.
Sept 21
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 28)

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.

The first midterm exam will be Tuesday, Sept 25 from 6:45–8:00pm.
See here for complete details, including how to register for the conflict exam.

Week 5

Sept 24
More on the gradient (§14.6) and overview of optimization (§14.7–14.8). Notes.
HW 12 (Due Mon, Oct 1)

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

Sept 25
Section: Review for midterm.
Midterm the First: 6:45–8:00pm. Solutions.
Sept 26
No class.
Sept 27
Midterm discussed. Solutions.
Sept 28
Local min and max (§14.7). Notes.
HW 13 (Due Wed, Oct 3)

2018: This was fine when I skipped any detailed discussion of why x^3 is unstable.

Different tutoring and office hours this week.

Week 6

Oct 1
Absolute min and max (§14.7). Notes.
Visualization: Min/max example.
HW 14 (Due Fri, Oct 5)

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 at this point.

Oct 2
Taylor series, the second derivative test, and changing coordinates. Worksheet. Solutions.
Oct 3
Constrained min/max (§14.8). Notes.
HW 15 (Due Mon, Oct 8)

2016 and 2018: 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.

In future, replace Problem 3 of HW 15 of something that doesn't use multiple constraints. (You can solve it without this, but many students don't notice this.)

Oct 4
Constrained min/max via Lagrange multipliers. Worksheet. Solutions.
Oct 5
Introduction to space curves (§13.1–4). Notes.
Visualization: Cycloid.
HW 16 (Due Wed, Oct 10)

2016: This lecture is ok, I got though everything though I wished it was a bit shorter. 2018: This was fine.

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.

The final exam for Math 241 will be Saturday, Dec 15 from 8–11 am

Week 7

Oct 8
More on arc length (§13.3) and integrating functions on curves (§16.2, pages 1063–1065). Notes.
HW 17 (Due Fri, Oct 12)

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: This was fine. I needed about 16 minutes to do everything after "Understanding these integrals"; in the second hour I did it in 13 which was kinda tight.

Oct 9
Lagrange multipliers problem discussion. Quiz on HW 11–15.

Exam 2 draft due (Patrick and James)

Oct 10
Vector fields (§16.1) and integrating them along curves (§16.2). Notes.
Example vector field: Live wind map
HW 18 (Due Mon, Oct 15)

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.

Oct 11
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 12
More on integrating vector fields along curves; the Fundamental Theorem of Line Integrals (§16.2 and §16.3). Notes.
HW 19 (Due Wed, Oct 17)

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.

The final exam for Math 241 will be Saturday, Dec 15 from 8–11 am

Week 8

Oct 15
Conservative vector fields I (§16.3). Notes.
HW 20 (Due Fri, Oct 19)

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.

Oct 16
Integrating vector fields. Worksheet. Solutions.

Exam 2 to printer.

Oct 17
Conservative vector fields II (§16.3). Notes.
HW 21 (Due Mon, Oct 22)

2016: 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. 2018: Similar.

Oct 18
Discussion of line integral problems. Quiz on HW 16–19. No office hour today.
Oct 19
Intro to multiple integrals (§15.1). Notes.
HW 22 (Due Fri, Oct 26)
Last day to drop the course (*)

2018: This is fine.

The second midterm exam will be Tuesday, Oct 23 from 6:45–8:00pm.
See here for complete details.

Tutoring room moved to 141 Altgeld for the rest of the semester, starting Oct 17.

(*) 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 22
Integrating over more complicated regions (§15.2 and §15.3). Notes.
HW 23 (Due Mon, Oct 29)

2016 and 2018: Got through notes as written.

Oct 23
Section: Review for midterm.
Midterm the Second: 6:45–8:00pm. Solutions.
Oct 24
No class.
Oct 25
Multivariable integrals. Worksheet. Solutions.
Oct 26
Polar coordinates (§15.3) and applications (§15.4). Notes.
HW 24 (Due Wed, Oct 31)

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: 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.

Different tutoring and office hours this week. Tutoring room is now 141 Altgeld.

Week 10

Oct 29
Triple integrals (§15.6). Notes.
HW 25 (Due Fri, Nov 2)

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.

Oct 30
Transformations of the plane. Worksheet. Solutions.
Oct 31
Integrating in cylindrical and spherical coordinates (§15.7 and §15.8). Notes.
HW 26 (Due Mon, Nov 5)

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.

Nov 1
Discussion of multivariable integral problems. Quiz on HW 22–24.
Nov 2
Changing coordinates I (§15.9). Notes.
HW 27 (Due Wed, Nov 7)

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

Week 11

Nov 5
Changing coordinates II (§15.9). Notes.
HW 28 (Due Fri, Nov 9)

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: 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 6
Integrating by changing coordinates. Worksheet. Solutions.

Exam 3 draft due (Nathan and Patrick)

Nov 7
Surfaces in R3 (§16.6). Notes.
HW 29 (Due Mon, Nov 12)

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

Nov 8
Surface Parameterpolooza. Worksheet. Solutions.
Nov 9
Area and integration on surfaces (§16.6 and §16.7). Notes.
Visualization: Cones.
HW 30 (Due Wed, Nov 14)

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 12
Green’s Theorem (§16.4). Notes.
HW 31 (Due Fri, Nov 16)

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

Nov 13
Parametrizations and integrals. Worksheet. Solutions.

Exam 3 to printer.

Nov 14
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 26)

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 15
Green's Theorem. Worksheet. Solutions.
Nov 16
The Divergence Theorem in 2D (§16.5-16.9). Notes.
HW 33 (Due Fri, Nov 30)

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.

The third midterm exam will be Tuesday, Nov 27 from 6:45–8:00pm.
See here for complete details.

Week 13

Nov 26
Surface integrals of vector fields and the divergence theorem in 3D (§16.7 and §16.9). Notes.
HW 34 (Due Mon, Dec 3)

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) so that the divergence theorem tells us we are computing the volume.

Nov 27
Section: Review for Midterm 3.
Midterm the Third: 6:45–8:00pm. Solutions.
Nov 28
No class
Nov 29
Midterm discussed. Solutions.
Nov 30
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 Patrick and James?)

Different tutoring and office hours this week.

Week 14

Dec 3
Stokes Theorem (§16.8), including the definition of the curl (§16.5). Notes.
HW 35 (Due Fri, Dec 7).

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

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.

Dec 4
Surface integrals of vector fields. Worksheet. Solutions.
Dec 5
More on Stokes Theorem (§16.8), including understanding the curl (§16.5). Review of conservative vector fields. Notes.
HW 36 (Due Mon, Dec 10)

This lecture is a good length.

Dec 6
Stokes’ Theorem. Worksheet. Solutions.

This worksheet was split in half from the corresponding one in 2016.

Final exam to printers

Dec 7
Conservative vector fields in R3 (§16.8); Topology 101. Notes.
HW 37 (Due Wed, Dec 12)

This lecture was a fine length.

Week 15

Dec 10
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.

Dec 11
More on Stokes’ Theorem. Worksheet. Solutions.

This is the second half of the Stokes' theorem worksheet from 2016.

Final exam to printers

Dec 12
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.

Dec 13
Reading day.
Dec 14
Finals begin.
Dec 15
Final the Ultimate will be held Saturday, December 15, from 8–11am. Solutions.
Extra office hours and tutoring later in the week, see here for complete details.

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