Math 223 - Multivariable Calculus
Spring 2025
Emily Proctor




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

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

And here is an extended problem list. This list includes all of our assigned problems, along with some extras. If you are looking for more practice, the extras are good ones to work with.




Homework

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Week beginning February 10

Due Wednesday, February 12:

• 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. (Update: please see my email and remove 28 from the list.)


• Write up and turn in the above problems at the start of class on Wednesday. You can do them on paper, or 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.


• Read Sections 1.3 and 1.4.

Due Friday, February 14:

• Here are the notes from class today.


• From Section 1.3 do problems: 3, 9, 15, 22, 24, 25, 29, 37.


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


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


• Happy Valentine's Day. :)

Week beginning February 17

Due Monday, February 17:

• In class, I talked about the parametric equations of a plane. With this in mind, it might help you to go back take a look again at Problem 1.1.22.


• From Section 1.5 do problems: 1, 4, 8, 11, 12, 14, 16, 19, 22, 23, (25)*.


• *Problem 25 is optional and ungraded. It's a distance problem, which we didn't cover in class. I mention this problem to give you a reason to read Examples 7, 8, and 9 in the book as a way to deepen your understanding of projections and dot products.


• From Section 1.6 do problems: 10*, 11*.


• *Problems 10 and 11 are a lot like Problem 1.3.37. You could use the solution to that problem as inspiration for these two.


• Read Section 1.7.

Due Wednesday, February 19:

• From Section 1.7 do problems: 9, 11, 15, 17, 20, 23, 24a, 30, 33, 36, 37, 42.


• Read Section 2.1, p.86-95

Due Friday, February 21:

• From Section 1.7, do problems: 18, 27, 46ab*.


• *For Problem 46, 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.


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


• *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.


• Read Section 2.1, p.96-99.

Week beginning February 24

Due Monday, February 24:

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


• *Problems 32, 33, and 37a 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.97.


• Here is the chart of quadric surfaces that we looked at in class today. It might be helpful as you consider Problems 40-42. As always, if you can't get the picture to look good, just include a written description as well.


• Flip through Section 2.2. This is a relatively long section, which we will treat fairly lightly in class. That said, the author does a good job of explaining multidimensional limits. We will not cover the official definition of the limit explicitly in our class, but if you are curious about it (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.

Due Wednesday, February 26:

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


• *For Problem 13, try simplifying.


• **For Problems 33 and 35, see Examples 8, 9, and 10.


• ***For problems 39 and 45, briefly justify your answer.


• Read Section 2.3, p.120-122 and Section 2.4, p.140-142.

Due Friday, February 28:

• From Section 2.3 do problems: 2, 3, 5, 8, 15.


• From Section 2.4 do problems: 14, 24abcd, 25, 30.


• Read Section 2.3, p.122-127.

Week beginning March 3

Due Monday, March 3:

• From Section 2.3, do problems: 39b, 40, 41, 42, 44, 47*.


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


• *Try out Problem 47, noticing the definition of DF above Definition 3.8. This problem is a bridge between what we are covered in class on Friday and what we will be learning about on Monday.


• Start preparing for our exam on Monday, March 10, 7-9pm. It will cover from the beginning of the semester through Section 2.4. Here is a list of topics to help you prepare for the exam.


• Read Section 2.3, p.127-132 and Section 2.4, p.138-140.

Due Wednesday, March 5:

• From Section 2.3 do problems: 28, 32, 35, 37, 45, 46, 63.


• Answer Question 1 on the following total derivative worksheet. We will be doing the worksheet in class, so have a copy of your answers ready to work with in class. (If you like, since this is a big concept, you could also take a peak ahead at the rest of the worksheet and/or the videos to start to see in advance how it all comes together.)


• Here is a sample exam and solutions for you to practice with. You can also prepare one side of a 4x6 index card of notes to bring to the exam with you.

Due Friday, March 7:

• There is no homework due. If you are looking for more practice with the total derivative, I recommend Problems 48 and 49 in Section 2.3.


• On Wednesday, we interpreted of the derivative in a bit more depth than the book does. We'll be thinking this way later on in the semester. Since there are no problems due, it'd probably be worth it to use the time to look again at the reading/notes/videos to help this material sink in.


• Read Section 2.5


• Continue to prepare for our exam on Monday, March 10, 7-9pm, in Warner 104 and 105. The exam will cover from the beginning of the semester through Section 2.4. Here again is a list of topics that will show up on the exam. Remember that you can bring one side of a 4x6 index card with notes to the exam with you.

Week beginning March 10

Due Monday, March 10:

• Continue to prepare for the exam on Monday night. The exam will cover up through Section 2.4 and will take place in Warner 104 and 105.


• 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 12:

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


• Read Section 2.6 p.158-163.

Due Friday, March 14:

• From Section 2.6 do problems: 1, 2, 3, 6, 9ab, 11, 12, 15.


• Read Section 3.1.

Week beginning March 17

Happy spring break!

Week beginning March 24

Due Monday, March 24:

• From Section 3.1 do problems: 3 (just do 0 to 2pi), 5, 14, 15b*, 21 (give a vector equation), 29**, 31, 33***, 34.


• *For Problem 15b, 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 29.


• ***In Problem 33, note that if ||x(t)|| is constant, then so is ||x(t)||^2. Problem 31 might be of help here.


• Read Section 2.6 p. 172-175.


• In order to make Monday's class 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.


• 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 26:

• From Section 2.6 do problems: 20, 25, 27, 28, 31 (just do method b), 36, 38.


• Read Section 3.2 p.210-213.

Due Friday, March 28:

• 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.


• Read Section 3.3 p.229-232 and Section 3.4 p.236-242.

Week beginning March 31

Due Monday, March 31:

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


• From Section 3.4 do problems: 4, 10, 13, 16, 17, 18*, 25, 33**.


• *The phrase "f and g are functions of class C^2" in Problem 3.4.18 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 3.4.33, 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.


• Read Section 4.1 p.254-266 and Section 4.2 p.273-278.

Due Wednesday, April 2:

• From Section 4.1 do problems: 5, 21, 24*, 25*, 27.


• *For Problems 24 and 25, multiply out the matrices so that you arrive at an actual polynomial.


• Read Section 4.2 p.277-285.

Due Friday, April 4:

• From Section 4.2 do problems: 3, 8, 11, 22a, 29*, 44.


• *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.278 of the book is a more general version (for functions R^n to R) than the version we did in class. For the specific version we did in class (for functions R^2 to R), see Example 5 on p.279. 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.278).


• I did not give the full proof of the second derivative test for functions R^2 to R in class, but if you are curious about it, here are some 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.


• Here is a write-up of the example that we did at the end of class, which highlights a method for figuring out the critical points of a function.


• Read Section 4.3 p.289-295.


• In order to get more out of class on Friday, it might be helpful to review how gradients and level sets are related (specifically Theorem 6.4 p.172) before watching.

Week beginning April 7

Due Monday, April 7:

• 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.


• Related: I talked about the Extreme Value Theorem at the end of class, and showed how it applied in our example. Go back and have a look at p.281-285 in Section 4.2 to learn more about it, and how it can (and cannot) be applied to find extreme values. Note that the method in Example 9 on p.283 appeared in the book before Lagrange multipliers. For the boundary in Problem 23, try Lagrange multipliers instead!


• Start preparing for our exam on Monday, April 14, 7-9pm. The exam will cover Section 2.5 (Chain Rule) through Section 5.2, 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 class, in the homework, or on the list above, it won't be on the test.


• Read Sections 5.1 and 5.2 p.326-333.

Due Wednesday, April 9:

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


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


• Read Section 5.2 p.333-340.


• Here is a sample exam and solutions for you to practice with. You can also prepare one side of a 4x6 index card of notes to bring to the exam with you.

Due Friday, April 11:

• Nothing! I hope you will go check out the Spring Symposium!

Week beginning April 14

Due Monday, April 14:

• To practice for the exam, from Section 5.2 do problems: 3a, 9, 21, 24, 28*, 38**. You do not need to turn these problems in! Here are solutions so that you can check your work.


• *For Problem 5.2.28, 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 5.2.38, just set it up. You do not need to carry out the integral.


• Continue to prepare for our exam on Monday, April 14, 7-9pm, in Warner 104 and 105. The exam will cover Section 2.5 through Section 5.2. Here again is a list of topics that will show up on the exam. Remember that you can bring one side of a 4x6 index card with notes to the exam with you.


• 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, April 16:

• From Section 5.3 try problem: 1. You do not need to turn it in! We'll go over it in class on Wednesday.


• Read Section 5.3 and Section 5.4.

Due Friday, April 18:

• 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).


• Read or review Section 5.4.

Week beginning April 21

Due Monday, April 21:

• 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!


• Read Section 5.5 p.366-368, 379-381.


• 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.

Due Wednesday, April 23:

• From Section 5.5 do problem: 7*.


• From the following change of variables worksheet do problems: 1-7. Look at and consider (possibly even carry out, if they make sense to you) the rest of the problems on the worksheet as well. They will be the foundation of class on Wednesday.


• *For Problem 5.5.7, for each part of the problem, the word "determine" means draw a picture (and/or describe). 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.


• Read Read Section 5.5, p.377-379.

Due Friday, April 25:

• From our change of variables worksheet do problems: 8-11.


• From Section 5.5 do problems: 9*, 14, 16, 20**, 21***, 27, 28.


• *For Problem 5.5.9, see Example 9, p.374. Note that the problem gives you u and v in terms of x and y, but to use what we learned in class, you'll need to rearrange to get x and y in terms of u and v. Drawing a picture of the region should help you determine the range of values for u and v.


• **Problem 5.5.20 is a little different from the others because the upper limit for r depends on theta. To figure out the relationship between r and theta, try converting x^2 + (y-1)^1=1 into polar coordinates.


• ***For Problem 5.5.21, try taking the difference between a rectangular integral and a polar integral.


• Read Section 5.5, p.382-388.

Week beginning April 28

Due Monday, April 28:

• From Section 5.5 do problems: 26, 30, 31, 34, 36, 39*, 43**.


• *For Problem 39, just set up the integral but don't compute. Which coordinate system should you use?


• **For Problem 43, this is a stretch problem. Consider your coordinate system carefully, and try using a different order of integration than we typically use.


• Read Section 6.1, p.431-435.