4th Q Projects

Below are ideas for your fourth quarter projects. Feel free to choose one and expand on it, or pick something on your own and (through consultation with me) run with it! The fundamental thing you need to think about when choosing a topic is: does this excite me? If it doesn’t, if you don’t get the sense of “oh my gosh, I can’t wait to start working on this,” you should think twice about choosing it.

(1) The Volume of the Sphere in Higher Dimensions. In 2D, a sphere can be described by $x^2+y^2 \leq r^2$. In 3D, a sphere can be described by $x^2+y^2+z^2 \leq r^2$. Well, we can talk about a sphere in $n$-dimensions by defining it like $x_1^2+x_2^2+...+x_n^2 \leq r^2$. Using what we know about multivariable calculus, believe it or not, it is relatively easy to calculate the volume of an $n$-dimensional sphere. It turns out that the volume of a 5th dimensional sphere of radius 1 will be a maximum, and then the volume of 6th, 7th, 8th, … dimensional spheres will be less. In fact, as the dimension increases, the volume gets closer and closer to 0. This is weird — seriously weird. Explore this idea. Perhaps if you can, expand this idea by calculating the surface area of spheres in $n$-dimensions. Resource: Rogawski, Section 15.3.

UPDATE: a student did this in the 2009/10 class (but not in super detail, so it could be fleshed out)

UPDATE: a student did this in the 2012/13 class extensively

(2) Hurricane Modeling: In higher mathematics, you can take courses on “fluid dynamics” — how fluids move. In such a course, you’ll learn that gases can be model as fluids, and some of the beasts we’ve encountered in Multivariable Calculus will rear their heads again (e.g. vector valued functions, vector fields, etc.). Anton explores how one can model hurricanes using what we’ve learned thus far (p. 1183-1187). Read the text and do the 12 exercises presented. If time permits, do some basic research on “fluid dynamics.”

(3) Lissajous Curves: Earlier in this course, we’ve encountered Lissajous curves. They were pretty, but we never really explored them. They can be (in 2D) described by the parametric equations $x(t)=A\cos(\omega_x t-\delta_x)$ and $y(t)=B\cos(\omega_y t-\delta_y)$. Do some research on these curves and write up a paper explaining the properties of these curves based on the equations that produce them. Perhaps think about them in 3D — as we did in class. (4) Harmonograph: If you are interested in Lissajous curves but are good at creating things, build a “harmonograph” — a machine that produces Lissajous Curves. Watch the video below to see what I’m talking about:

It appears to involve a few dowels, some clamps, some weights, and a lot of patience.

UPDATE: A student did this in the 2008/9 class, and a video of his harmonograph is below

(5) If you learned Newton’s Method (an elegant algorithm to find the zeros of a function in one dimensions), do some research on Newton’s Method in 2 dimensions. In other words, find at what points $f_1(x,y)=0$ and $f_2(x,y)=0$ (when two equations are both equal to zero).
Resources:
http://www.math.gatech.edu/~carlen/2507/notes/NewtonMethod.html http://ktuce.ktu.edu.tr/~pehlivan/numerical_analysis/chap01/NewtonSystem.pdf

(6) River Navigation Problem: I found this problem (and solution) online. If you chose to do this, you must promise not to look at the solution. You can come to me when you get stuck, and we can work through those tough patches together. However, I want you to work on this without peeking at the solution.

Kirk and Rose have misread their maps while on a hiking date. After walking around aimlessly for three hours they finally come upon a ranger station. It is on the opposite side of a river whose straight banks are 150 feet apart. The good news is that there is a small motor boat tied to a dock with a call box on their side of the river. They phone the ranger station and learn: (1) the river current flows parallel to its banks, and its speed at any pointxfeet from either bank is $6\sin(\frac{\pi}{150}x)$ feet per second; (2) the boat travels at a constant rate of 5 feet per second in still water. They also receive some bad news: (1) the boat has a broken rudder and can only head in a fixed direction; (2) the land is infested with poisonous snakes on the ranger station side of the river, so they must head off at an angle that results in their arriving exactly at the dock in front of the ranger station. So at what angle should they set off to arrive at the ranger station.

(7) Explore the challenge problems listed at Oliver Knill’s Multivariable Calculus website. Each of the problems are given a ranking (*** or ****) based on difficulty. Do enough problems so that you have amassed 20 *s.

(8) As you’ve noticed, much of what we’ve done in Multivariable Calculus is based in vectors. But where did this come from? Who did this come from? Read Michael J. Crowe’s book A History of Vector Analysis: The Evolution of the Idea of the Vectoral System (preview of book here at Google) and write a comprehensive book review and a deep historical analysis.

UPDATE: a student did this in the 2013/14 class

(9) Contour Maps and Gradients: There are lots of map materials available on the web (EPA, US Geological Survey). The use of global positioning systems (GPS) is leading to tremendously accurate maps. Examine level sets of various properties (rainfall, temperature, etc). Calculate directional derivatives, gradients, draw flow lines, calculate average value of functions all based on these maps. This project was directly lifted from here.

(10) Maps: Maps of the world and the different projections used to draw them. The problem of how to represent a three dimensional sphere on a two dimensional paper is tricky and has been approached in many different ways. One standard method is the Mercantor projection. This topic relates to different coordinate systems: note that spherical coordinates are closely related to longitude and latitude. Like the previous project, this project was directly lifted from here.

(11) Tools for the Mathematics Teacher. Much of what we’ve covered this year is visual. WinPlot has been extraordinarily useful for us. However, there are times when WinPlot just doesn’t cut it. For example, when explaining double integrals, Fred’s cookie cutter analogy couldn’t be beat! Dream up some tools that would make explaining some concepts from Multivariable Calculus easier. For example, you can create a set of wire sculptures that illustrate curvature/radius of curvature. If you know how to work with clay, you can create some famous surfaces. You can come up with a foam model that shows students the idea of double integrals, or triple integrals. You could create some beautiful sliceforms. Or use string to show the intersection of various surfaces, a la Theodore Olivier (see some info and pictures of his models here). I have some photographs of some of Olivier’s models from when I visited the Musee des Arts et Metiers in Paris, so if you want to see them, I can pass them along for your edification and inspiration. Resource: http://mathforum.org/models/choices.html

(12) William Thomson (a.k.a. Lord Kelvin) studied — among many other things — bubbles. He said “Blow a soap bubble and observe it. You may study it all your life and draw one lesson after another in physics from it.” Investigate the mathematics behind soap bubbles and their relationship to “minimal surfaces” — surfaces whose mean curvature is zero (see Wikipedia page here). Create these surfaces by making wire frames from hangers and dipping them into a soap solution. Do experiments and take photographs of your bubbles, write a paper, go in whatever direction you want!
Resources:
http://epinet.anu.edu.au/mathematics/minimal_surfaces
http://www.instructables.com/id/Minimal-Surfaces-With-Metal-Shapes-and-Soap-Bubble/ http://www.math.northwestern.edu/~wphooper/fun/bubbles/

(13) Centroid and Center of Gravity. Get some foam board and draw an accurate coordinate system on it. Accurately draw a few regions on the board and calculate the centroid of this figure. Label the centroid on the figure. To show it truly is a centroid, carefully and safely use an exacto knife to accurately cut out the figure. Thread a needle and push it through the calculated centroid and and dangle the figure in the air. Does it rest flat or lean?

When you’ve finished that, use a second piece of foam board with an accurate (rectangular) grid on it. Measure the weight of the board, and use that to calculate the weight of each square drawn on it. Then stick some heavy pushpins in the center of a random selection of squares. Measure the weight of a single pushpin. Using this information, calculate the center of gravity of this object. Thread a needle and push it through the calculated center of gravity. Does it rest flat or lean?

Lastly, cut out a few random squares from that second piece of foam board, and repeat.

An example of this, forming mathematical art, is here. Perhaps this could be expanded by designing mobiles that accurately hang in balance?

UPDATE: A student did this in the 2008/9 class.

UPDATE: A student did the 3-D version of this in the 2009/10 class.

(14) Research Maxwell’s Equations and explain them. Resource: Daniel Fleisch’s A Student’s Guide to Maxwell’s Equations.

UPDATE: A student did this in the 2008/9 class.

UPDATE: A student did this in the 2012/13 class.

(15) Investigate the Ideal Gas Law ( $PV=nRT$) focusing on what the 3D plotting of P vs. V vs. T looks like, given constants n and R. (You can use this Excel Spreadsheet to help you.) Find the shape of isotherms (level curves of equal temperature). Then investigate the van Der Waal’s equation for real gases ( $(P+\frac{a}{V^2})(V-b)=RT$), where $a$ and $b$ are constants based on the specific gas considered. It turns out that the critical point(s) of this surface can inform us about the state (liquid, vapor) of the gas (see diagram below). Again, find the shape of isotherms. In your investigations, discuss the differences between the Ideal Gas Law and the van Der Waal’s equation for real gases. Calculate the critical point(s) for some real gases mathematically and explain what it/they mean. Resources:

Click to access Lecture5.pdf

http://demonstrations.wolfram.com/VanDerWaalsIsothermsForRealAndIdealGases/http://www.bpreid.com/applets/pvDemo.html

UPDATE: a student did this in the 2013/14 class

(16) If you know a thing or two about programming or are willing to learn some basic Mathematica, explore the Wolfram Demonstration Projects for Multivariable Calculus here. Find a topic in Multivariable Calculus that hasn’t been illustrated yet, or one that might be poorly illustrated, and create your own Demonstration! (If you are dedicated to doing this, please inform me ASAP because I have to (somehow) get you a copy of Mathematica. To convince yourself that it isn’t impossible and that the coding isn’t terribly long, please look at some of the source codes for the demonstrations you like (you can see the source code for every Demonstration by clicking on the button saying “Show Source Code” at the bottom of each Demonstration page).

(17) In class this year, we encountered how to find a normal vector (N) and a tangent vector (T) to a curve defined parametrically. We also briefly talked about and calculated a binormal vector (B) — a vector orthogonal to both the normal and tangent vector. It turns out that these three unit vectors (N, T, B) taken together form something called a “Frenet Frame.” These three variables are also related via curvature and torsion through a set of three equations known as the Frenet-Serret formulas (see Wikipedia page here). Investigate what torsion is (we have already talked about curvature), and write a paper for someone just slightly less capable in math than you are explaining the N, T and B are, what the Frenet-Serret formulas are, and if you have time, how this all relates to physics. Resource: http://demonstrations.wolfram.com/FrenetFrame/

(18) We have been learning to use WinPlot as we’ve been going along. Do some research (there are about a zillion webpages online for tutorials/instructions) on how to use WinPlot and create a student tutorial handbook (either a paper handbook, or even better, an online handbook) for next year’s Multivariable Calculus class for how to use WinPlot. It should cover how to use WinPlot basically (how to put gridlines on a graph, how to vary the window size, etc.) and how to do Multivariable Calculus specific things (e.g. plot in 3D, calculate the intersection of two surfaces, create level curves, etc.). WinPlot is powerful and this project will show you how powerful it truly is. (Plus the skills you learn here might help you in college.) If you are scared of making an online handbook, I can show you how. It is supremely easy! Resource: http://math.exeter.edu/rparris/wpsupp.html

[Update: Many of you have been using Mac’s Grapher… A similar thing can be done for Grapher.]

(19) Origami has actually started making headway into the lives of professional mathematicians — not only as a lark, but as a source of some deep mathematics (see the work of Eric Demaine, for example). In Thomas Hull’s book Project Origami, it appears that there is a project called “Five Intersecting Tetrahedra.” You can construct this figure and work on the mathematics behind this figure (Activity #11) in the book — where you will find the “optimal strut width” of this figure using multivariable calculus.

UPDATE: A student did this in the 2010/11 class.

(20) If you are taking AP Statistics and enjoyed the treatment of probability, you can investigate the relationship between calculus (and multivariable calculus) and probability theory. (We already used multivariable calculus to come up with the area under the normal curve and prove that the equations for the line of best fit is indeed true.) Think of a probability density function $\phi(x)$  a function which gives you the probability of an event happening (namely, latex $x$). You already know that $\int_{-\infty}^{+\infty} \phi(x)dx=1$ — the “sum” of the probability of all possible outcomes has to be 1. Investigate probability density functions, their means, their variances, and other basic statistical concepts — but using calculus (or if needed, multivariable calculus). Resource: http://www.math.montana.edu/frankw/ccp/multiworld/topic.htm#multiple

(21) Here are pictures of a wine holder, with and without a wine bottle in it.   Find a wine holder like it and analyze it mathematically, to calculate (a) how it will look with empty bottles of various lengths and (b)  how it will look with a bottle filled with different amounts of wine. Then design your own cool wine holder and analyze it.

UPDATE: a student did this in the 2014/15 class

(22) The following game-theory puzzle lifted from here has a fascinating solution:

Alice, Bob, and Charlie are chosen to participate in an auction for an economics class.

Each submits a secret bid of 1, 2, or 3. The person with the lowest andunique* bid wins the auction and gets a prize of \$1.

*Example: If Alice and Bob bid 1, and Charlie bids 2, then Charlie wins. Although Alice and Bob bid lower numbers, their bids were not unique so they do not win.

(Also: if they all bid the same number, no one wins)

If each plays the same mixed strategy, what is the optimal strategy? (what is the symmetric mixed Nash equilibrium?)

Fair warning: this game is harder to analyze than it sounds!

Understand this problem, try to come up with the solution without looking at the website, and then come up with a similar but different problem and solve it.

UPDATE: A student did this in the 2012/13 class.

(23) Although not technically in the realm of multivariable calculus, you can investigate fractional derivatives. Thus far, you’ve learned about 1st, 2nd, 3rd, etc., derivatives. But what about generalizing the idea of a derivative so you can have a fractional derivative, like derivative 1/2. What the what?! Investigate fractional calculus.

(24) A rabbit is running around. It is pursued by a wolf, which always runs in the direction of the rabbit. Analyze the situation. The path of the wolf is called a pursuit curve and the math behind them is elegant, and the drawings of various pursuit curves can be beautiful. UPDATE: a student did this in the 2013/14 class

(25) In calculus, you learned briefly about differential equations. There is a multivariable version of differential equations which incorporates multiple space variables and a time variable, and leads to some interesting physics (the wave equation, the heat equation, etc.). Some examples of the equations and how they allow a surface to change in time are here. Learn a little bit about partial differential equations, and then pick one or two equations (I can help you with this) and study ’em in detail.

(26) Watch this video:

Create one of these yourselves at school! Here are two pages that will help with this (one, two). You is a possibility that you might have to purchase some of the materials yourselves, but there could be some sources of funding that we could tap into, as well as forage some things (like a projector) at school!

(27) This is a magic square (made from legos). It balances at the center. Turns out, they all do. Prove this fact algebraically, explain it conceptually, build one (or two… or three…), and then come up with an extension project that builds off of what you’ve learned. (28) I found this Reddit thread which talks about going from conic sections (which are slices of a cone) to quadric surfaces… The question is: Is there a 4D “cone” which these quadric surfaces are all slices of? What are the features/properties of this 4D “cone”? What does it “look” like? Read over the thread, tinker and play around with the ideas mentioned, and create a project based on this question or some related question.

(29) A student asked me “If you have a surface and two points, is there a way to find the path of shortest length that connects the two points on the surface?” On a plane, this is a line. On a sphere, this would be… (think about it). And what about one some crazy surface? I didn’t know the answer — so I did some googling and found the notion of the “geodesic.” The math might get a bit ugly and beyond us, but if you’re interested, do some research to try to understand the ideas behind finding geodesics! And do some calculations for some simple surfaces? [FYI: This ties in to general relativity!]

(30) When we read Steven Strogatz’s The Calculus of Friendship, he mentioned the brachistochrone (which is also a tautochrone). You can analyze this problem analytically (on paper) understanding the mathematics and physics behind it… but then you can actually build a brachistochrone and show that it is a tautochrone by racing marbles down it! (Here is an awesome youtube video on the brachistochrone by one of my favorite math youtube-rs and Steven Strogatz! And here is another one by Vsauce and the Mythbusters guy!)

(31) Other ideas can be found here. Also, you can consider designing your own 3 day unit (with lesson plans, homework, smartboards, worksheets, etc.) for a particular topic that we learned that captured your interest.

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27 07 2011

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