Multivariable Calculus

Multivariable Calculus Projects 2015-2016

Each year, I have students in my multivariable calculus class do “fourth quarter projects.” We continue working with the material during classtime, they have regular nightly work, but I cancel all problem sets and tests. Instead, students choose a project topic they are interested in pursuing that has some relationship to the course (even if the relationship is a bit tenuous). I want the project to be one of passion. Their entire fourth quarter grade is based on these projects. This year, my kids came up with some amazing projects — some of the best I’ve seen in my eight years of teaching this course. (Some previous years projects are here, here, here, and here.)

An Augmented Reality Sandbox


Earlier in the year, I showed my student a video of an augmented reality sandbox that I stumbled across online. She showed interested in making it. It takes in a mapping of a surface (in this case sand in a sandbox) and projects onto the surface colors representing the height of the sand over time (so red is “high” and blue is “low”). The cool part about this is that the projection changes live — so if you change the sand height, the projection updates with new colors. Level curves are also “drawn” on the sand.

Here are some videos of it in action (apologies for the music… I had to put music on it so the conversations happening during the playing with the sand were drowned out):

The student was going to design lesson plans around this to highlight concepts in multivariable calculus (directional derivative, gradients, gradient field, reading contour maps) but ran out of time. However upon my suggestion, during her presentation, she did give students contour maps of surfaces, turned off the projector, had students try to form the sand so it matched the contour map, and then turned the projector on to have students see if they were right or not.

During the presentation, one student who I taught last year (but not this year) said: “This is the coolest thing I’ve seen all year!” and then when playing with the sand: “I AM A GOD!” Entrancing!


In my first year of teaching this course, a student was entranced by lissajous curves when we encountered them. These are simple parametric equations which create beautiful graphs. I then suggested for his final project that he create a harmonograph, which he did. Seven years later, I had another student see the original video of my student’s harmonograph, and he wanted to build his own! But he wanted his to have a rotary component, in addition to two pendulums which swung laterally. So he found instructions online and built it!


Here are some of the images it produced:

And here is a video of the harmonograph in motion:

(You can watch another video here.)


During the presentation, the student talked about the damping effect, how the pendulum amplitudes and periods had an effect on the outcome, and how lissajous curves were simply shadows of lissajous knots that exist in 3-space. Because of the presentation, I had some insights into these curves that I hadn’t had before! (I still don’t know how mathematically to account for how the rotary pendulum in the student’s harmonograph affects the equations… I do know that it has the harmonograph — in essence — graph the lissajous curves on a somewhat rotating sphere (instead of a flat plane). And that’s interesting!

Teaching Devices for Multivariable Calculus

A student was interested in creating tools for teachers to illustrate “big” multivariable calculus ideas… Contour lines, directional derivatives, double integrals, etc. So she made a set of five of super awesome teaching manipulatives.  Here are three of them.

The first is a strange shaped cutout of poster-cardboard-ish material, with four animals hanging from it. Then there is string connected to a magnet on top, and another magnet on the bottom. If you hold up the string and you aren’t at the center of mass, the mobile won’t balance. But if you move the magnet around (and the student used felt around the magnet so it moves seamlessly!), you can change the position of the string, until it balances. This is a manipulative to talk about center of mass/torque.


Another is a set of figures that form “level curves.” At first I was skeptical. The student said the manipulative elow was to help students understand countour plots. I wanted to know how… Then the moment of genius…


You can change the height of the level curves to make the “hill” steeper and steeper, and then look straight down at the manipulative. If you have a shallow “hill,” you have contour lines which will look far apart. If you have a tall “hill,” you have contour lines which look close together.

Finally, a third manipulative showcases the tangent plane (and it can move around the surface because of magnets also). I can see this also being useful for normal vectors and even surface integrals!


tangent plane.png

Cartographic Mapping

Two students decided to work together on a project dealing with cartographic mapping. They were intrigued by the idea that the surface of the earth can’t perfectly be represented on a flat plane. (They had to learn about why — a theorem by Euler in 1777.) They chose two projections: the Gall Peters projection and the Stereographic projection.

They did a fantastic job of showing and explaining the equations for these projections — and in their paper, they went into even more depth (talking about the Jacobian!). It was marvelous. But they had two more surprises. They used the 3D printer (something I know nothing about, but I told them that they might want to consider using to to create a model to illustrate their projections to their audience) and in two different live demos, showed how these projections work. I didn’t get good pictures, but I did take a video after the fact showing the stereographic projection in action. Notice at the end, all the squares have equal area, but the quadrilaterals on the surface most definitely do not have equal area.

An added bonus, which actually turned out to be a huge part of their project, was writing an extensive paper on the history of cartography, and a critical analysis of the uses of cartography. They concluded by stating:

We have attempted, in this paper, to provide our readers with a brief historical overview of cartography and its biases.  This paper is also an attempt to impress upon the reader the subjective nature of a deeply mathematical endeavor.  While most maps are based around mathematical projections, this does not exclude them from carrying biases.  In fact, we believe there is no separation between mathematical applications and subjectivity; one cannot divorce math from perspective nor maps from their biases.  We believe it is important to incorporate reflections such as this one into any mathematical study.  It is dangerous to believe in the objectivity of scientific and numerical thought and in the separation between the user and her objective tools, because it vests us, mathematicians and scientists, with arbitrary power to claim Truth where there is only perspective.

Beautiful. And well-evidenced.

Deriving the Hagen-Poiseuille Equation from the Navier-Stokes Equations

One student was interested in fluid dynamics. So I introduced him to the Navier Stokes equations, and set him loose. This turned out to be a challenging project for the student because most of the texts out there require a high level of understanding. Even when I looked at my fluid dynamics book from college when I was giving it to him as a reference, I realized following most of it would be almost impossible. As he worked through the terms and equations, he found a perfect entree. He learned about an equation that predicts the change in pressure from one end of a tube of small radius to another (if the fluid flow in the tube is laminar). And so using all he had learned in his investigation of the field, he could actually understand and explain algebraically and conceptually how the derivation worked. Some of his slides…


It was beautiful because he got to learn about partial differential equations, and ton of ideas in fluid dynamics (viscosity, pressure, rotational velocity, sheer, laminar flow, turbulence, etc.), but even needed to calculate a double integral in cylindrical coordinates in his derivation!

The Wave Equation and Schrodinger’s Equation

This student works in a lab for his science research class — and the lab does something with lasers and quantum tunneling. But the student didn’t know the math behind quantum mechanics. So he spent a lot of time working to understand the wave equation, and then some time trying to understand the parts of Schrodinger’s equation.

In his paper, he derived the wave equation. And then he applied his understanding of the wave equation to a particular problem:


He then tackled Schrodinger’s Wave Equation and saw how energy is quantized! Most importantly, how the math suggests that! I remember wondering how in the world we could ever go from continuousness to discreteness, and this was the type of problem where I was like “WHOA!” I’m glad he could see that too! Part of this derivation is below.


Overall, I was blown away by the creativity and deep thinking that went into these final projects. Most significantly, I need to emphasize that I can’t take credit for them. I was incredibly hands off. My standard practice involves: having students submit three ideas, I sit down with students and help them — with my understanding of their topics and what’s doable versus not doable — narrow it down to a single topic. Students submit a prospectus and timeline. Then I let them go running. I don’t even do regular formal check-ins (there are too many of them for me to do that). So I have them see me if they need help, are stuck, need guidance or motivation, whatever. I met with most of them once or twice, but that’s about it. This is all them. I wish I could claim credit, but I can’t. I just got out of their way and let them figure things out.

Pitching college math courses

Ooops. This turned out to be a post with no images. So here’s a TL;DR to whet your appetite: I wanted to expose my seniors to what college mathematics is, but instead of lecturing, I had them “pitch” a college course to the rest of the class.

My multivariable calculus courses was coming to an end, and I got some questions about what college courses in math are about. It reminded me of a comic strip I read years ago, which I frustratingly can’t find again. It has an undergraduate going to meet with his math professor adviser, saying something like “I want to major in triple integrals.” Which is crazy-sounding — but maybe not to a high school student who has only ever seen math as a path that culminates in calculus. What more is out there? What is higher level math about? (These questions are related to this post I wrote.)

So here’s what I told my students to do. They were asked to go onto their future college math department websites (or course catalog), scour the course offerings, and find 3-4 courses that looked interesting and throw these courses down on a google doc.

It was awesome, and made me jealous that they had the opportunities to take all these awesome classes. Some examples?


After looking through all the courses, I highlighted one per student that seemed like it involved topics that other students had also chosen — but so that all the courses were different branches/types of math. I told each student to spend 10-15 minutes researching their highlighted course — looking up what the words meant, what the big ideas were, finding interesting videos that might illustrate the ideas — so they can “pitch the course to the class” (read: explain what cool math is involved to make others want to take the course).

I’m fairly certain my kids spent more than 10-15 minutes researching the courses (I’m glad!). Each day, I reserved time for 2-3 students to “pitch” their courses. And since some of the ideas were beyond them, after the pitches, I would spend 5 or so minutes giving examples or elaborating on some of the ideas they covered.

If you want to see the research they did for their pitches, the google doc they chucked their information into is here.

Some fun things we did during the pitches?

(1) We watched a short clip of a video about how to solve the heat equation (that was for a course in partial differential equations)

(2) I showed students how to turn a communication network into a matrix, and explained the meaning of squaring or cubing the matrix (this was for a course on network theory)

(3) A student had us play games on a torus (a maze, tic tac toe) (this was for a course on topology)

(4) I had students store x=0.3 on their calculators. Then I had each student store a different “r” value (carefully chosen by me) and then type r*x*(1-x)->x in their calculators. They then pressed enter a lot of times. (In other words, they were iterating x_{n+1}=rx_n(1-x_n) with the same initial conditions but slightly different systems. Some students, depending on their r value, saw after a while their x values settle down. Some had x values that bounced between two values. Some had x values that bounced between four values. And one had x values that never seemed to settle down. In other words, I introduced them to a simple system with wacky wacky outcomes! (If you don’t know about it, try it!) (This was for a course on chaos theory)

(5) A student introduced us to Godel’s incompleteness theorem and the halting problem (through a youtube video)

It was good fun. It was an “on the spot” idea that turned out to work. I think it was because students were genuinely interested in the courses they chose! If I taught a course like AP Calculus, I could see myself doing something similar. I’m not sure how I would adapt this for other classes… I’m thinking of my 9th grade Advanced Geometry class… I could see doing something similar with them. In fact, it would be a great idea because then they could start getting a sense of some of the big ideas in non-high school mathematics. Kay, my brain is whirring. Must stop now.

If anyone knows of a great and fun introduction to the branches of college level math (or big questions of research/investigation), I’d love to know about it. Something like this is fine, but it doesn’t get me excited about the math. I want something that makes me ooh and ahh and say “These are great avenues of inquiry! I want to do all of them!” I think those things that elicit oohs and ahhs might be the paradoxes, the unintuitive results, the beautiful images, the powerful applications, the open questions… If none exists, maybe we can crowdsource a google doc which can do this…

Reading? For math class?

This year, our school adopted this weird rotating schedule where we see our classes 5 times out of every 7 days. And four of those times are 50 minute classes and one of those times is a 90 minute class.

I didn’t have a clear idea of what to do in multivariable calculus for the block. I still had to cover content, but I wanted it to be “different” also. After many hours of brainstorming, I came up with a solution that has worked out pretty well this year.

We had a book club.

The 90 minute block was divided into 50 minutes of traditional class, and 40 minutes of book club. (Or 60 minutes of class, and 30 minutes of book club.)

Now, to be clear, this is a class of seven seniors who are highly motivated and interested in mathematics. I can see ways to adapt it in a more limited way to other courses, with more students, but this post is about my class this year.


We started out reading Edwin Abbott’s Flatland.


Why? Because after they read this, they understand why I can’t help them visualize the fourth (spatial) dimension! But it convinces them that they can still understand what it is (by analogy) and makes them agree: if we can believe in the first, second, and third spatial dimensions, why wouldn’t we believe in higher spatial dimensions too? It’s more ludicrous not to believe they exist than to believe they don’t exist! A perfect entree into multivariable calculus, wouldn’t you say?

After reading this, we read the article “The Paradox of Proof” by Caroline Chen on the proposed solution to the ABC conjecture.


This led us to the notion of “modern mathematics” (mathematics is not just done by dead white guys) and raised interesting questions of fairness, and what it means to be part of a profession. Does being a mathematician come with responsibilities? What does clear writing have to do with mathematics? (Which helps me justify all the writing I ask for on their problem sets!) It also started to raise deep philosophical questions about mathematical Truth and whether it exists external to the human mind. (If someone claims a proof but no one verifies it, is it True? If someone claims a proof and fifty people verify it, is it True? When do we get Truth? Is it ever attainable? Are we certain that 2+2=4?)

At this point, I wanted us to read a book that continued on with the themes of the course – implicitly, if not explicitly. So we read Steve Strogatz’s The Calculus of Friendship:


What was extra cool is that Steve agreed to sign and inscribe the book to my kids! The book involves a decades long correspondence between Steve and one of his high school math teachers. There are wonderful calculus tricks and beautiful problems with explanations intertwined with a very human story about a young man who was finding his way. Struggling with choosing a major in college. Feelings of pride and inadequacy. The kids found a lot to latch onto both emotionally and mathematically. Two things: we learned and practiced “differentiating under the integral sign” (a Feynman trick) and talked about the complex relationship that exists between teachers and students.

After students finished this book, I had each student write a letter to the author. I gave very little guidelines, but I figured the book is all about letters, so it would be fitting to have my kids write letters to Steve! (And I mailed the letters to Steve, of course, who graciously wrote the class a letter back in return.)

Our penultimate reading was G.H. Hardy’s A Mathematician’s Apology:


I went back and forth about this reading, but I figured it is such a classic, why not? It turned out to be a perfect foil to Strogatz’s book — especially in terms of the authorial voice. (Hardy often sounds like a pompous jerk.)  It even brought up some of the ideas in the “Paradox of Proof” article. What is a mathematician’s purpose? What are the responsibilities of a mathematician? Why does one do mathematics? And for kids, it really raised questions about how math can be “beautiful.” How can we talk about something that is seen as Objective and Distant to be “beautiful”? What does beauty even mean? Every section in this essay raises points of discussion, whether it be clarification or points that students are ready to debate.

What is perfect about this reading is at the same time we were doing it, the movie about G.H. Hardy and S. Ramanujan was released: The Man Who Knew Infinity (based on the book of the same name).

Finally, we read half of Edward Frenkel’s Love and Math:


Why? Because I wanted my students to see what a modern mathematician does. That the landscape of modern mathematics isn’t what they have seen in high school, but so much bigger, with grand questions. And through Frenkel’s engaging telling of his life starting in the oppressive Russia and ending up in the United States, and his desire to describe the Langland’s program understandably to the reader, I figured we’d get doses of both what modern mathematics looks like, and simultaneously, how the pursuit of mathematics is a fully human endeavor, constrained by social circumstances, with ups and downs. Theorems do not come out of nowhere.Mathematicians aren’t the blurbs we read in the textbooks. They are so much more. (Sadly, we didn’t read the whole thing because the year came to a close too quickly.)


I broke the books into smaller chunks and assigned only them. For Flatland, it might have been 20-30 pages. For Love and Math or A Mathematician’s Apology, it might have been 30-50 pages. We have our long block every 7 school days, so that’s how much time they had to read the text.

At the start, with Flatland, students were simply asked to do the reading. Two students were assigned to be “leaders” who were to come in with a set of discussions ready, maybe an activity based on something they read. And they led, while I intervened as necessary.

For every book club, students who weren’t leading were asked to bring food and drink for the class, and we had a nice and relaxing time. On that note,  never did I mention anything about grades. Or that they were being graded during book club. (And they weren’t.) It was done purely for fun.

Later in the year, I had students each come to class with 3-4 discussion questions prepared, and one person was asked to lead after everyone read their questions aloud.

The discussions were usually moderated by students, but I — depending on how the moderation was going — would jump in. There were numerous times I had to hold back sharing my thoughts even though I desperately wanted to concur or disagree with a statement a student had made. And to be fair, there were numerous times when I should have held back before throwing my two cents in. But my main intervention was getting kids to go back to the texts. If they made a claim that was textually based, I would have them find where and we’d all turn there.

Sometimes the conversations veered away from the texts. Often. But it was because students were wondering about something, or had a larger philosophical point to make (“Is math created or discovered?”) which was prompted by something they read. And most of the times, to keep the relaxed atmosphere and let student interest to guide the conversation, I allowed it. But every so often I would jump in because we had strayed so far that I felt we weren’t doing the text we had read justice (and we needed to honor that) or we were just getting to vague/general/abstract to say anything useful.


I mentioned students generated discussion questions on their own. Here are some, randomly chosen, to share:

  •  Strogatz talks about how math is a very social activity. We see this exemplified in the letters between Steve and Mr. Joffray, but where else do we see this exemplified in math? (papers, etc.) How do you think Strogatz might have felt about Shinichi Mochizuki’s unwillingness to explain his paper and proof to the math community?
  • What do you think about Strogatz and Joff using computer programs to give answers to their problems? Are computers props, and their answers unsatisfying? Or are they just another method, like Feynman’s differentiating under the integral?
  • Do you like A Square? In what ways is he a product of his society? Does he earn any redeeming qualities by the end of the book?
  • Can you draw any connections between things in Flatland and religion? Do you think Abbott is religious? Why/why not?
  • When we first read about Mochizuki’s ABC Conjecture, we debated whether or not math is a “social” subject. Perhaps many mathematicians do much of the “grind” work on their own, however, throughout everything we’ve read this year, there has been one common link when it comes to the social aspects of math: mentorship. It appears to me that all of the great mathematicians we know about have been mentored by, or were mentors others. In what ways have Frenkel’s mentors – he’s had a few – had an influence on the path of his mathematical career? Do you think he would/could be where he is today without all of those people along the way? Can you think of any mentors that have had a profound influence on your life? (The last one can just be a thought, not a share.)
  • Frenkel talks about the way in which math, particularly interpretations of space and higher dimensions, began to influence other sectors of society, specifically the cubist movement in modern art. This movement was certainly not the first time math and science influenced art and culture – think about the advent of perspective in the Renaissance and the use of technology on modern art now – however math and art are often thought as opposites and highly incompatible. Why do you think that people rarely associate the two subjects? Would you agree that the two are incompatible? Can you think of other examples of math/science influence art/culture/society?


In many ways, I felt like this was a perfect way to use 30 minutes of the long block. After doing it for the year, there are a few things that stood out to me, that I want to record before summer hits and I forget:

(a) I think students really enjoyed. It isn’t only a vague impression, but when I gave a written survey to the class to take the temperature of things, quite a few kids noted how much they are enjoying the book clubs.

(b) For the post-Flatland book club meetings, I need to come up with multiple “structures” to vary what the meetings look like. Right now they are: everyone reads their discussion questions, the leader looks for where to start the discussion, the discussion happens. But I wonder if there aren’t other ways to go about things.

One example  I was thinking was students write (beforehand) their discussion questions beforehand on posterpaper and bring it to class. We hang them up, and students silently walk around the room writing responses and thoughts on the whiteboard. Then we start having a discussion.

Or we break into smaller groups and have specific discussions (that I or students have preplanned) and then present the main points of the discussion to the entire class.

Clearly, I need to get some ideas from English teachers. :)

(c) I love close readings of texts. I think it shows focus, and calls on tough critical thinking skills. At the same time, I need to remember that this is not what the book clubs are fundamentally about. They are — at the heart, for me — inspiration for kids. So although for Flatland I need to keep the critical thinking skills and close readings happening, I need to remember (like I did this year) to keep things informal.

(d) Fairly frequently, I will know something that is relevant to the conversation. For example, I might talk about of the math ideas that were going over their heads, or about fin de siecle Vienna, or branches of math that might show how the line between “theoretical” and “applied” math is blurry at best. I have to remember to be judicious about what I talk about, when, and why. We only have limited time in book club, so a five minute tangent is significant. And one thing I could try out is jot down notes each time I want to talk about something, and then at the end of the book club (or the beginning of the next class), I could say them all at once.

(e) I usually reserve 30 minutes for book club. But truthfully, for most, 40 minutes turned out to be necessary. So I have to keep that in mind next year when planning class.

(f) Should we come up with collaborative book club norms? Should I have formal training on how to be a book club leader? Should we give feedback to the leaders after each book club? Can we get the space to feel “safe” where feedback could actually work?

And… that’s all!

Multivariable Calculus Projects 2014-2015

At the end of each year in Multivariable Calculus, I have students develop and execute their own “final project.” It’s fairly open-ended and students end up finding something they are personally interested/invested in and they go for it.

This year I had six students and these are their projects.

“Exploring the Normal Distribution Through the Box-Muller Transform and Visualizing It Using Computer Science” (GT)


This student had never taken a statistics course but was interested in that. We also talked about how to find the area under the normal distribution using multivariable calculus (and showed it was 1). Armed with those two things, this student who likes computer science found a way to pick independently two numbers (one each from two uniform distributions), and have them undergo a few transformations involving square roots and sine/cosines, and then those two numbers would generate two new numbers. Doing this a bunch of times will create a whole pile of new numbers, and it turns out that those square roots and sine/cosines somehow create a bunch of numbers that exactly follow a normal distribution. So weird. So cool.

“XRayField: Detecting Minecraft Cheating using Physics and Calculus” (W.M.)

diamondThis student loves Minecraft and hosts a Minecraft server where tons of kids at our school play. Earlier in the year, there was a big scandal because there were people cheating when playing on this server — using modifications to give themselves additional advantages. (This was even chronicled by the school newspaper.) One of the modifications allows players to see where the diamonds are hidden, so they can dig right to them. So this student who runs the server wanted to find a way to detect cheaters. So he created a force field around each diamond (using the inverse square law in 3D), and then essentially calculated the work done by the force field on the motion of a player. A player moving directly with the force field (like on the left in the image above) will get a higher “work score” than someone on the right (which is moving sometimes with the forcefield, sometimes not). In other words, he’s calculating a line integral in a field. His data was impressive. He had some students cheat to see what would happen, and others not. And in this process, he even caught a cheater who had been cheating undetected. Honestly, this might be one of my favorite projects of all time because of how unique it was, and how perfectly it fit in with the course.

“Space Filling Curves” (L.S.)


This student with a more artistic bent was interested by “Space Filling Curves” (we saw some of them when I started talking about parametric curves in three dimensions, and we fiddled around with Lissajous curves to end up with some space filling curves). This student created three art pieces. The first was a 2D Hilbert curve which is space filling. The second was a 3D Hilbert curve which is space filling (pictured above). The third was writing a computer program to actually generate (live) a space filling curve which involves a parametrically defined curve, where each of the x(t) and y(t) equations involved an infinite sum (where each term in this infinite sum was reliant on this other weird piecewise and periodic function). I wish I had a video showing this program execute in real time, and how it graphed for us — live — a curve which was drawing itself and how that curve being drawn truly filled space. It blew my mind.

“The Math Is Right: The Math Behind Game Shows” (J.S.)


This student, since a young age, loved watching the Game Show Network with his mother. So for his final project, he wanted to analyze game shows — specifically Deal or No Deal, and the big wheel in the Price is Right. I had never thought deeply about the mathematics of both, but he addressed the question: “When should you take the deal? Is there an optimal time to do so?” (with Deal or No Deal) and “If you’re the second player spinning the big wheel (out of three players), how do you decide whether to spin a second time or not?” (for the Price is Right). As I saw him work through this project — especially the Price is Right problem — I saw so much rich mathematics unfold, involving generating functions, combining distributions, and simulating. It’s a deceptively simple question, with a beautifully rich analysis that hides behind it. And that can be extended in so many ways.

“The Art of Balance” (M.S.)


This photograph may make it look like the books are touching the wine holder. That is not the case. This wine holder is standing up — quite robustly as we tested — through it’s own volition. And — importantly — because the student who built it understood the principle behind the center of mass. This student’s project started out with him analyzing the “book stacking problem” (which involves how much “overhang” you can create while stacking books at the edge of the table. For example, with one book, you can put it halfway over the table and it will not fall. It turns out that you can actually get infinite overhang… you just need a lot of books. This analysis centered around the center of mass of these books, and actually had this student construct a giant tower of books. The second part of this project involved the creation of this wine holder, which was initially conceived of mathematically using center of mass, then that got complicated so the student started playing around with torque which got more complicated, so the student eventually used intuition and guess and check (based on his general understanding of center of mass). Finally he got it to work. The one thing this student wanted to do for his project was “build/create something” and he did!

“Visualizing Calculus” (T.J.)


This student wanted to make visualizations of some of the things we’ve learned about this year. So he took it upon himself to learn some of the code needed to make Wolfram Demonstrations, and then went forth to do it. He first was fascinated with the idea of fractional derivatives, so he made a visualization of that. Then he wanted to illustrate the idea of the gradient and how the gradient of a 2D surface in 3D space sort of defined a plane tangent to the surface if you zoomed in enough. Finally, he created an applet where the user enters a 2D vector field, and then it calculates the divergence and curl at every point of the vector field. His description for what the divergence was was interesting, and new to me. About the point chosen on the applet, he drew a circle (and the vector field was illustrated in the background). He said “imagine you have a light sprinkling of sand on this whole x-y plane… and then wind started pushing it around — where the wind is represented by the vector field, so the direction and strength of the wind is determined by the vector field. If more sand is coming into the circle and leaving it, then the divergence is negative, if more sand is leaving the circle than coming into it, then the divergence is positive, and if equal amounts of sand are coming in and leaving the circle, then the divergence is zero.”

Multivariable Calculus Final Projects 2013-2014

Instead of doing traditional problem sets and tests during the fourth quarter, I have kids work on an individual project on something that relates to multivariable calculus that interests them. (During the year, I have them keep track of interesting tidbits or facts or something I go off on a tangent about [pun] that they find could be a possible final project. I also have this list of ideas I’ve culled to help them come up with a topic.)

I have them come up with a prospectus and I individually talk with kids about their proposed project and timeline for completion. Then when they get started and start envisioning a final product, they are asked to write a description of the final product out clearly, and come up with a rubric for grading that product. They are also asked to make a 20-25 minute presentation to their classmates, their parents (if they choose to invite them), math teachers, and administrators. This year, they wanted to give their presentations during senior thesis week, which means that lots of their friends could come to their talks.

And they have been! In the past week, students have given their talks and I have been way impressed by them. Honestly they’ve been more independent than in years’s past, so I was unsure of whether they were putting together a solid final project or not. They did.

Without further ado:

Title: Mathematical Change We Can Believe In
Description: This presentation shows how one region can be manipulated to form something more interesting, a process called Transformation of Axes. The 2D and 3D analogues, use of rectangular and rounded shapes, and proofs of the properties of transformations abound in this exciting journey through the wonders of the world of multiple (MANY) variables.


Title: Pursuit Curves: The Ultimate Game of Tag
Description: Pursuit curves are the paths formed when one point chases another point. In this program, we will be looking at the mathematical explanations of pursuit curves, and then using a computer program I have built to model a few.


Title: What’s Our Vector, Victor
Description: This will be an investigation into the history, origins, and evolution of vectors, their analysis, and notation.


Title: Economists working with Models: Understanding the Utility Function
Description: Firstly, we will gain a foundational understanding of economics as a discipline. Secondly we will discuss the utility function and the questions which it raises.


Title: From Chemistry to Calculus: a study of gas laws
Description: For my project I have constructed a “textbook” that analyzes the idael and real gas law through the lens of multivariable calculus. In my “textbook” I compare and constrast these two laws by means of graphical and derivative analysis.


Title: Knot Theory
Description: Knots are everywhere around us, from how we tie our shoes to how the proteins in our body wind themselves up. My presentation will give an overview of their place not only in the “real world,” but also the world of classroom math and calculus.


Green’s Theorem and Polygons

Two nights ago, I assigned my multivariable calculus class a problem from our textbook (Anton, Section 15.4, Problem 38). Even though I’ve stopped using Anton for my non-AP Calculus class, I have found that Anton does a good job treating the multivariable calculus material. I think the problems are quite nice.

Anyway, the problem was in the section on Green’s theorem, and stated:

(a) Let C be the line segment from a point (a,b) to a point (c,d). Show that:

\int_C -y\text{ }dx+x\text{ }dy=ad-bc

(b) Use the result in part (a) to show that the area A of a triangle with successive vertices (x_1,y_1),\text{ }(x_2,y_2), and (x_3,y_3) going counterclockwise is:


(c) Find a formula for the area of a polygon with successive vertices (x_1,y_1),\text{ }(x_2,y_2),...,(x_n,y_n) going counter-clockwise.

Today we started talking about our solutions. We all were fine with part (a). But part (b) was the exciting part, because of the variation in approaches. We had five different ways we were able to get the area of the triangle.

  • There was the expected way, which one student got using part (a). This was the way the book intended the students to solve the problem — and I checked using the solution manual to confirm this. What was awesome was that even though we as a class understood the algebra behind this answer, a student still asked for a conceptualgeometric understanding of what the heck that line integral really meant. I knew the answer, but I left it as an exercise for the class to think about. So we’re not done with this problem.

  • There was a way where a student made a drawing of an arbitrary triangle and then used three line integrals of the form \int_C y\text{ }dx to solve it. In essence, this student was taking the area of a large trapezoid (calculated by using a line integral) and subtracting out the area of two smaller trapezoids (again calculated by using line integrals). Another student astutely pointed out that even though we had an arbitrary triangle, the way we set up the integral was based on the way we drew the triangle — and to be general, we’d have to draw all possibilities. You don’t need to understand precisely what this means — because I know I”m not being clear. The point is, we had a short discussion about what would need to be done to actually have a rigorous proof.

  • There was a way where a student translated the triangle so that the three vertices weren’t (x_1,y_1),\text{ }(x_2,y_2),\text{ }(x_3,y_3) anymore… but instead (0,0),\text{ }(x_2-x_1,y_2-y_1),\text{ }(x_3-x_1,y_3-y_1). Then he used something we proved earlier, that the area of a triangle defined by the origin and two points would involve a simple determinant (divided by 2). And when he did this, he got the right answer.

  • Another two students drew the triangle, put it in a rectangle, and then calculated the area of the triangle by breaking up the rectangle into pieces and subtracting out all parts of the rectangle that weren’t in the triangle. A simple geometric method.

  • My solution involved noticing that \frac{1}{2}(ad-bc) is the area of a triangle with vertices (0,0),\text{ }(a,b),\text{ }(c,d). And so I constructed a solution where a triangle is the sum of the areas of two larger triangles, but then with subtracting out another triangle.

The point of this isn’t to share with you the solutions themselves, or how to solve the problem. The point is to say: I really liked this problem because it generated so many different approaches. We ended up spending pretty much the whole period discussing it and it’s varied forms (when I had only planned 10 or 15 minutes for it). I liked how these kids made a connection between a previous problem we had solved (#28) and used that to undergird their conceptual understanding. I loved how these approaches gave rise to some awesome questions — including “what the heck is the physical interpretation of that line integral in part (a)?” In fact, at the end of class, we were drawing on paper, tearing areas apart, trying to make sense of that line integral. All because a student suggested that’s what we do. (Again, I have made sense of it… but I wanted the kids to go through the sense making process themselves… their weekend work is to understanding the meaning of this line integral.)

I don’t know the real point of posting this — except that I wanted to archive this unexpectedly rich problem. Because it’s not that it is algebraically intensive (though some approaches did get algebraically intensive). Rather, it’s because it is conceptually deep.

MV Calculus Projects 2010-2011

One class that I think I am pretty free about, and we have some fun in and get to explore and go through a lot of productive frustration, is my multivariable calculus class. I had 5 students in it last year. (*As he ducks from the rotten vegetables hurled his way, and collective groan from the crowd.*) Sadly, next year, there will be no students eligible so I won’t be teaching it.

My favorite thing from this course is the fourth quarter projects that all students do. We don’t have problem sets, we don’t have any tests or quizzes. Just this thing.

At the beginning of the year, I tell students to write down random things that pique their interests, whet their appetites, for the fourth quarter project. Whether it be higher dimensions, to the notion of curvature and what that might mean for surfaces, to the use of optimization problems in various fields, to whatever. As the class goes on, I’ll mention some interesting tidbit here or there and sometimes they’ll add it to the list in the back of their notebook. And then comes the fourth quarter, where they basically get to pick anything they want, they write their own project prospectus, they write their own rubric, and they just go at it. I give them some options, but they don’t always go that way.

I meet with them once a week or every other week (or more if they need it) and provide guidance and support, sometimes needed, sometimes not.

This year I had some excellent projects. I can’t believe I didn’t outline them for you when we finished the year, so I will outline them now. What was great is that some parents got to come to the final presentations, and so did my department head, the head up of the upper school, and some math teachers. Different days had different audiences.

1. The first project involved constructing 5 intersecting tetrahedra out of origami and figuring out the “optimal strut width” (the width of the “beam” of each edge of the tetrahedra) so the tetrahedra just sit beautifully within each other without having them wiggle around (too small) or bend to fit together (too large).

This problem involves multivariable calculus, believe it or not, but also involved some really beautiful precalculus work meshed with 3D (basically, using roots of unity and some right triangle trigonometry) to find the vertices of a dodecahedron.  I also have to say that making the darn thing was totally hellish and the student who did it is a super rockstar. She also wrote a really comprehensive final paper explaining the calculations. Color me impressed.

2. Another student, who is a nationally recognized runner, wanted to investigate the following question: if you have a random surface with a local maximum, and you put yourself on that surface, and you wanted to get to the maximum, how would you get there? Instead of taking the shortest path (which would follow the gradient), the student conjectured that if you ran along the least steep path you will run faster, and if you run along the most steep path you will run slower. So there is a tradeoff, and there will be a path to run in between those two choices which will be optimal. So the student and I constructed a function to model the velocity of this runner. Although together we couldn’t actually get a general answer, or even a specific answer for a specific surface and point we chose, we had fun struggling through it. The student also created an accurate model of a one surface that the runner would be running on (the one that he did his calculations for).

3. Another student, for an earlier problem set where they were asked to write their own problems, studied the idea of marginal utility in economics and related that to Lagrange multipliers. This student was one of those kids who is interested in everything and he really loved studying marginal utility, and wanted to extend it and see how else multivariable calculus was used in economics. So he pretty much devoured this book on his own. Although he didn’t find too much multivariable calculus, he became enamored with the idea of the utility function, and decided to make a 50 minute class lesson on economics and calculus with an emphasis on the utility function. It was so well thought out, and so well delivered, that I think that teaching and simplifying ideas might be this kid’s calling. He also wrote an amazing paper outlining everything from the presentation (and more that he couldn’t fit in), and a problem set for students to work on after the presentation.

4. Say you have a blob drawn on graph paper, and you wanted to measure the area. What if I said: there is a mechanical device that if you drag it along the perimeter of the blob, it would calculate and tell you the area? True story, this exists, and when I described this to a student struggling to find a project… a project he was insistent he wanted to make with his hands… he was hooked. The device is called a planimeter. It sort of makes sense that something like this could exist… I mean:

(that’s Green’s theorem). So this mechanically minded student first built a trial version of a planimeter, using pencils, binder clips, and a bottle cap. And it worked fairly well. So then he built a giant and much more sturdy one. You can see this student holding his “draft” version and on the table is his professional version.

This student did almost all the work without me (which is good because I have no idea how to work with things mechanically). I basically only helped him understand some of the math behind how the mechanical device worked. The end result was that the professional device worked fairly well, but I think given another week, it could have been tenfold more accurate. Time is always the sticking point with these end of year presentations.

5. The final project was one of my favorites, because it involved me really going back and learning some simple partial differential equations. How this project happened involved me showing this student the following video:

Of course this video can’t but help stir the imagination. So this student wanted to build the device (called a Chaldni plate) and study the math behind it. It turned out that building the device was a bit beyond our capabilities, so we enlisted the help of the science department chair who super generously ordered a chaldni plate (he had the driver already) and helped get him set that up. I, on the other hand, did some research on what causes those beautiful patterns. Together, that student and I spent hours upon hours tearing through a paper — me doing a little lecture, him reading and asking questions, and so on and so on. And at the end, this student wrote his own paper based on our reading — explaining the math behind the designs. Although I don’t think he fully understood everything (we had not nearly enough time to make that possible), I loved that he got a touch of all these small things in higher level math. Orthogonal functions and Fourier series. 2D and 3D waves. Boundary conditions and time-dependent partial differential questions.

And his Chaldni plate worked.

PS. Apparently, I didn’t do a good job of blogging about my projects from previous years. Two years ago, here is what my kids did. And last year, I didn’t really write about it. Yikes! One student did a wonderful investigation on higher spatial dimensions, and how to extend what we’ve done into them — focusing on actually visualizing these dimensions (she really really really wanted to see them). The other extended a 2D project on center of mass that someone worked on the previous year, and I wrote about it obliquely here.