Math 223 - Multivariable Calculus
Spring 2024
Emily Proctor




Please read the syllabus to learn all the details about the course.

Here is our class document about collaboration. Please go back every once in a while and reread, and if you would like to add anything to it, please let me know!

And here is a chronological list of the videos and notes for the semester.




Homework

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Week beginning: February 12

Due Wednesday, February 14:

• Please type (up to) a page about who you are as a mathematician (classes or experiences you've had, anything you've liked, anything you've disliked...) as well as your motivations for taking Multivariable Calculus this semester. Be honest; this is just a simple, informal assignment that will help me start to get to know you. If there is anything else you'd like me to know as we start the semester, please include that too. Print out two copies of your page and turn them in at the start of class on Wednesday.


• Read Sections 1.1 and 1.2 of our textbook. (Note: you won't be responsible for knowing about the symmetric form of a line in R^n.)


• Watch Sections 1.1 and 1.2 video: Part 1/notes.


• From Section 1.1 do problems: 7, 10, 15, 17, 21a, 22, 23.


• From Section 1.2 do problems: 9, 10, 11, 17, 19, 28, 30, 35, 43.


• Bring questions from the homework to class with you on Wednesday, and we will talk more about it then. Your homework (neatly written, edges cut off, and stapled) will be due in the folder outside my office door by 5pm on Wednesday.


• If you would like to do your homework on a tablet and print out your work in order to turn it in, that is a perfectly great alternative as well.


• Here is an expanded version of the notes that I presented in today's video, if you would like to take a look.

Due Friday, February 16:

• Read Sections 1.3 and 1.4.


• Watch Sections 1.3 and 1.4 videos Part 1, Part 2, Part 3, and notes.


• From Section 1.3 do problems: 3, 9, 15, 18, 20, 21, 25, 33.


• From Section 1.4 do problems: 5 (it's okay to compute just one way), 9, 13, 17, 25, 26.


• Happy Valentine's Day. :)

Week beginning February 19

Due Monday, February 19:

• Read Section 1.5 and Section 1.6 (p.48-51).


• Watch Sections 1.5 and 1.6 videos Part 1, Part 2, Part 3, and notes.


• In part of these videos, I to talk a bit about the parametric equations of a plane. With this in mind, it might help you to go back before watching and take a look again at Problem 1.1.22.


• The notes that are posted here are a slightly expanded version of the notes I used in the videos. These notes include proofs of the Cauchy-Schwarz inequality and the triangle inequality, in case you would like to see them.


• From Section 1.5 do problems: 1, 4, 8, 11, 12, 14, 16, 19, 22, 23, 25. (Problem 25 is a distance problem. Even though we didn't cover the method in class, I put this problem on the assignment to prompt you to read Examples 7, 8, and 9 in the book. The goal is to help you understand material about projections and dot products better, so reading these examples is as important as doing the problem.)


• From Section 1.6 do problems: 10, 11.

Due Wednesday, February 21:

• Read Section 1.7.


• Watch Section 1.7 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 1.7 do problems: 9, 11, 15, 18, 19, 20a, 23, 26, 29, 32, 33, 35, 38, 42ab*.


• *For Problem 42, make note of which method for describing the given region is simpler. Later on, when we are integrating in three dimensions, this type of thinking will help you to set up integrals so that they are as simple to compute as possible.

Due Friday, February 23:

• Read Section 2.1, p.82-91.


• Watch Section 2.1 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 2.1, do problems: 2abc, 5, 11, 13, 15, 17, 22, 29.


• Note: for Problems 15, 17, and 22, if you have a hard time drawing the graphs, include a little written description of what you mean it to look like.

Week beginning Monday, February 26

Due Monday, February 26:

• Read Section 2.1, p.92-95.


• Watch Section 2.1 videos Part 4/notes and Part 5/notes.


• From Section 2.1, do problems: 32*, 33, 37, 38**, 39, 40, 41, 42, 46.


• *Problems 32, 33, and 37 are just asking for some level surfaces. You do not need to put them together into a graph (it would be impossible!).


• **Problem 38 is highlighting an important concept. Pay attention to this one, in conjunction with today's video and the italicized comment in the middle of p.93.


• Here is the chart of quadric surfaces mentioned in today's video. It might be helpful as you consider Problems 40-42.


• If you looking to go back and practice some more with cylindrical and spherical coordinates, problems 1.7.23-35 would be good extra problems to play with. These are completely optional, to work on on your own, as much or as little as you like.

Due Wednesday, February 28:

• Read Section 2.2. This is a relatively long section, but the author does a good job of describing things. We will not cover it explicitly in our class, but if you are curious about the official definition of the limit (which is a major concept of math, and which shows up in later math classes), it may be fun to take some time to read this section through. You do not need to read the addendum at the end unless you are curious about that too.


• Watch Section 2.2 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 2.2, do problems: 7, 11, 12, 13*, 19, 23, 33, 35, 39**, 45**. 47.


• *For Problem 13, try simplifying.


• **Briefly justify your answers for problems 39 and 45.

Due Friday, March 1:

• Read Section 2.3, p.116-118 and Section 2.4, p.136-138.


• Watch Section 2.3 and 2.4 videos Part 1/notes and Part 2/notes.


• From Section 2.3 do problems: 2, 3, 5, 8, 13.


• From Section 2.4 do problems: 14, 22abcd, 23, 28.

Week beginning Monday, March 4

Due Monday, March 4:

• Read Section 2.3, p.118-123


• Watch Section 2.3 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 2.3, do problems: 35, 37, 40, 42, 45.


• In case it is helpful, here is an example of how to compute the equation for a tangent plane.


• Make note of Problem 45; it is a bridge between what we are covering here and what you will be learning about for Wednesday.


• Start preparing for our exam on Tuesday (March 12, 7-9pm) of next week. It will cover from the beginning of the semester through Friday's assignment this week, which will be on Section 2.5. Here is a list of topics to help you prepare for the exam.

Due Wednesday, March 6:

• Read Section 2.3, p.123-128. Read Section 2.4, p.133-135.


• Watch Section 2.3 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 2.3 do problems: 26, 30, 33, 44, 59.


• From Section 2.4 do problems: 2.


• The material from the end of the videos about the interpretation of the derivative goes into a bit more depth than the book does. We'll be thinking this way later on in the semester. Since there are fewer problems due than usual, it would be worth it to take another look or two at the reading/video/notes now to help that material sink in.

Due Friday, March 8:

• Read Section 2.5.


• Watch Section 2.5 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 2.5 do problems: 2, 3, 4, 8a, 28.


• Here is the total derivative worksheet from Wednesday's class if you would like to continue to consider and play with it more quietly on your own.


• Continue to prepare for our exam next Tuesday, March 12, 7-9pm, in Johnson 204. The exam will cover from the beginning of the semester through Section 2.5. Here again is a list of topics to help you review.


• Looking ahead, for Monday's class, we will have an optional review session. I won't come with an agenda, so perhaps start collecting up any questions you might have!

Week beginning March 11

Due Monday, March 11:

• Continue to prepare for the exam on Tuesday night. The exam will cover up through Section 2.5 and will take place in Johnson 204.


• Monday's class will be an optional review session. I won't come with an agenda, so please bring any questions you have!

Due Wednesday, March 13:

• Nothing! I hope you have a good rest after our exam.

Due Friday, March 15:

• Read Section 2.6 p.158-163.


• Watch Section 2.6 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 2.6 do problems: 1, 2, 3, 6, 9ab, 11, 12, 13.

Week beginning March 18

Happy spring break!

Week beginning March 25

Due Monday, March 25:

• Read Section 3.1.


• Watch Section 3.1 videos Part 1/notes and Part 2/notes.


• From Section 3.1 do problems: 3 (just do 0 to 2pi), 5, 10, 11b*, 17 (give a vector equation), 25**, 27, 29***, 30.


• *For Problem 11b, you do not need the picture from part a in order to do part b.


• **If you would like it, here is a description about how to think about Problem 25.


• ***In Problem 29, note that if ||x(t)|| is constant, then so is ||x(t)||^2. Problem 27 might be of help here.


• Take a few minutes to look over your exam, together with the solution sheet. Compare your answers and see if you have any lingering questions that you'd like to clear up before we keep moving forward in class. I'm happy to help!

Due Wednesday, March 27:

• Read Section 2.6 p. 164-168.


• Watch Section 2.6 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• In order to make today's video more meaningful, it'd be worth it to review the notion that every surface in R^3 can be thought of as the level surface of some function.


• From Section 2.6 do problems: 18, 23, 25, 26, 29 (just do method b), 36.

Due Friday, March 29:

• Read Section 3.2 p.203-205.


• Watch Section 3.2 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 3.2 do problems: 3, 5, 10, 11, 12ab, 14*, 15**.


• *For Problem 14, to compute an indefinite integral of f(t) from a to infinity, compute the definite integral from a to b, then take the limit as b goes to infinity.


• **In Problem 15, you'll need to use the polar conversion formulas x=rcos(theta), y=rsin(theta). Theta is the defining parameter here. This means that as theta increases, the distance from the point on the curve to the origin (i.e. r) changes, depending on theta. Thus, the curve goes counterclockwise around the origin, but moves closer or farther away from the origin as it goes.

Week beginning April 1

Due Monday, April 1:

• Read Section 3.3 p.221-224 and Section 3.4 p.227-232.


• Watch Sections 3.3 video Part 1/notes.


• Watch Section 3.4 videos Part 1/notes and Part 2/notes.


• From Section 3.3 do problems: 3, 24a.


• From Section 3.4 do problems: 4, 10, 13, 14, 15, 16*, 23, 31**.


• *The phrase "f and g are functions of class C^2" in Problem 3.4.16 means that f and g can both be differentiated two times and their second derivatives are continuous. It's a technical requirement that ensures that any appropriate theorems can be applied here.


• **For Problem 3.4.31, it might be helpful to go back and look in your notes at the place where we derived the formula D_uf=(grad f) dot u.


• Thank you for the feedback that you gave in class today. If you have any more feedback that you'd like to offer about our class so far, here is an anonymous form that you can use for that.

Due Wednesday, April 3:

• Read Section 4.1 p.244-256 and Section 4.2 p.263-267.


• Watch Section 4.1 videos Part 1/notes and Part 2/notes.


• Watch Section 4.2 video Part 1/notes.


• From Section 4.1 do problems: 5, 19, 22*, 23*, 25.


• *For Problems 22 and 23, multiply out the matrices so that you arrive at an actual polynomial.

Due Friday, April 5:

• Read Section 4.2 p.267-274.


• Watch Section 4.2 videos Part 2/notes and Part 3/notes.


• From Section 4.2 do problems: 3, 8, 11, 22a, 29*, 42.


• *For Problem 29, your work will be much easier if you minimize the *square* of the distance rather than the distance itself.


• The version of the second derivative test given on p.268 of the book is a more general version (for functions R^n to R) than the version I did in the video. For the specific version I did in the video (for functions R^2 to R), see Example 5 on p.269. For the second derivative test, we will only ever consider functions R^2 to R (i.e. you are not responsible for knowing the general version on p.268).


• I did not give the full proof of the second derivative test for functions R^2 to R in the video, but if you are curious about it, here are the notes I wrote up that give the proof. It's a really beautiful application of diagonalizability, which you learned about in linear algebra. So, it is not required but I still encourage you to take a look at the notes, both to see why the second derivative test works, and for more practice with reading proofs.


• I talked about the Extreme Value Theorem at the end of the video, but only talked through the strategy of an example. I didn't assign you any problems based on the Extreme Value Theorem, but take a look at the following details of the example, along with Examples 8 and 9 in the book to get an idea of how the theorem can and can't be used in determining the extreme values of a function. We will use the Extreme Value Theorem a bit in the next section, when we work on Lagrange multipliers.

Week beginning April 8

Due Monday, April 8:

• Read Section 4.3 p.278-284


• Watch Section 4.3 videos Part 1/notes and Part 2/notes.


• In order to get more out of the video for today, it might be helpful to review how gradients and level sets are related (specifically Theorem 6.4 p.164) before watching.


• From Section 4.3 do problems: 1*, 3, 7, 13a, 21, 23**.


• *For Problem 1, once again, remember that minimizing the square of the distance will make your work simpler!


• **For Problem 23, you are trying to find the maximum and minimum value of f over the entire closed disk (not just the boundary circle). To find critical points on the interior, you can use the method of finding critical points that we learned in Section 4.2. Since the boundary of the disk is a level set, you can find critical points on the boundary by using Lagrange multipliers. Once you have found all of the critical points, make an argument about which critical point(s) give(s) the absolute maximum and which critical point(s) give(s) the absolute minimum.


• Start preparing for our exam on Thursday (April 18, 7-9pm) of next week. The exam will cover Section 2.5 (Chain Rule) through Section 5.3, not including Sections 2.7 or 4.4. Here is a list of topics to help you prepare for the exam. There is a lot of material in Chapter 3 that we didn't do (e.g. Kepler's laws and differential geometry). If you didn't see a topic in the videos, in the homework, or on the list above, it won't be on the test.

Due Wednesday, April 10:

• Read Sections 5.1 and 5.2 p.314-321.


• Watch Section 5.1/5.2 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 5.1 do problems: 5, 7, 8, 10, 11, 16.


• From Section 5.2 do problems: 3b, 14.

Due Friday, April 12:

• Read Section 5.2 p.321-328.


• Watch Section 5.2 videos Part 1/notes, Part 2/notes, and Part 3/notes.


• From Section 5.2 do problems: 3a, 9, 19, 22, 26*, 36**.


• From Section 5.3 do problem: 1***.


• *For Problem 5.2.26, do part a. For part b, just set it up. (If you *were* to carry out the integral, which method of integration would you need to use?)


• **For Problem 5.2.36, just set it up. You do not need to carry out the integral.


• ***For Problem 5.3.1, try it out. We'll go over it in the video for Monday.


• Continue to prepare for our exam on Thursday (April 18, 7-9pm) of next week, which will take place in Johnson 204. The exam will cover Section 2.5 (Chain Rule) through Section 5.3, not including Sections 2.7 or 4.4. There's a lot of material in Chapter 3 that we didn't do (e.g. Kepler's laws and differential geometry). If you didn't see a topic in the videos or in the homework, it won't be on the test.

Week beginning April 15

Due Monday, April 15:

• Read Section 5.3 and Section 5.4.


• Watch Section 5.3 video Part 1/notes.


• Watch Section 5.4 videos Part 1/notes and Part 2/notes.


• From Section 5.3 do problems: 5*, 6*, 10**, 13**, 15.


• From Section 5.4 do problems: 2, 5, 11***, 16****.


• *For Problems 5.3.5 and 5.3.6, just compute the integral one time (you can choose which order you'd prefer).


• **For Problems 5.3.10 and 5.3.13, just change the order of integration, but don't evaluate.


• ***For Problem 5.4.11, the term "cylinder" refers to a surface in R^3 that arises from an equation where only two variables are present.


• ****For Problem 5.4.16, just set up the integral but don't evaluate (unless you would like to try it out!).


• When we do triple integrals, we'll be drawing the regions of integration. Now is a good time to go back and review graphing surfaces, with a focus on paraboloids, cones, ellipsoids, and cylinders (as above).


• Continue to prepare for our exam on Thursday, April 18, 7-9pm. The exam will cover Sections 2.5 through Section 5.3. Here, again, is a list of topics that will appear of the exam.

Due Wednesday, April 17:

• Continue to prepare for the exam on Thursday night. The exam will cover Section 2.5 through Section 5.3 and will take place in Johnson 204.


• Wednesday's class will be an optional review session. I won't come with an agenda, so please bring any questions you have!

Due Friday, April 19:

• No class Friday. I hope you have fun at the spring symposium!

Week beginning April 22

Due Monday, April 22:

• Watch Section 5.4 videos Part 3/notes and Part 4/notes.


• From Section 5.4 do problems: 20*, 21, 26**, 29abc.


• *For Problem 20, it is okay to just set up the integral but not solve it. That said, consider the region and function, and try to choose the order of integration that would be the most straightforward to carry out.


• **Although Problem 26 does not specifically ask you to draw the region of integration, please do.


• I did not assign Problem 25 but I recommend it if you would like more practice!

Due Wednesday, April 24:

• Read Section 5.5 p.349-351, 362-364.


• Watch Section 5.5 videos Part 1/notes, Part 2/notes, Part 3/notes, and Part 4/notes.


• From Section 5.5 do problem: 7*.


• Review our total derivative worksheet from earlier in the semester, and familiarize yourself with (or, even better (!), recompute) the answers to Problems 1, 2, and 3 from the worksheet. We will be building off of these problems in class, so please come to class having already done/reviewed them, with the answers in hand.


• *For Problem 7, for each part of the problem, please draw a picture of D. Observe that the transformation in Problem 7 is the spherical coordinate transformation.


• The main changes of variables that we will be using will be from rectangular into polar, cylindrical, or spherical coordinates. Now would be a great time to go back to Section 1.7 to refamiliarize yourself with these coordinate systems.

Due Friday, April 26:

• Read Section 5.5, p.360-362.


• Watch Section 5.5 video Part 5/notes.


• From Section 5.5 do problems: 9, 14, 16, 18*, 19, 25, 26.


• *Problem 18 is a little different from the others. We'll talk more about it during class time on Friday.


• In case you would like to continue to play with it, here is the change of variables worksheet from class on Wednesday.

Week beginning April 29

Due Monday, April 29:

• Read Section 5.5, p.364-371.


• Watch Section 5.5 videos Part 6/notes and Part 7/notes.


• From Section 5.5 do problems: 24, 28, 29, 32, 34, 37*, 41**.


• *For Problem 37, just set up the integral but don't compute. Which coordinate system should you use?


• **For Problem 41, consider your coordinate system carefully, and try using a different order of integration than we typically use.