I know it's common for high school teachers to use software (such as Geogebra) to formulate geometry problems for their students, so I wonder: Do professors of multivariable calculus use softwares (like MATLAB, Maple or any other that plot surfaces and curves) to formulate exercises and problems for their students? Can anyone give me examples?
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5$\begingroup$ Not the answer that you want, but, I find the standard examples are more than enough challenge for the students. It follows I can usually set-up whatever I need via algebra and what I know geometrically. Of course, students would be wise to use a CAS or 3D graphing utility where appropriate. $\endgroup$– James S. CookCommented Jul 18, 2017 at 0:58
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$\begingroup$ My students and I use web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D extensively when working on multivariable calculus questions. $\endgroup$– Steven GubkinCommented Jul 20, 2017 at 16:59
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$\begingroup$ You know the MITOCW platform? this is the multivariate calculus curse ocw.mit.edu/courses/mathematics/… $\endgroup$– BrainOntubeCommented Jul 23, 2017 at 21:40
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3 Answers
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There are already several "yes" answers so I'll give you a qualified "no":
What you do in class is what your students learn, so it depends on what you want to teach them. They should of course be aware the tools exist because it's hard to start learning how to visualize equations in 3d, but the more you as a teacher lean on them the less time they will have to see how to visualize equations without the help of the computer.
So why teach them visualize equations at all? In my opinion, the geometry skills we're building in Calc 3 are
- Looking at 2 to 4 variable equations and understanding their rough geometric structure and analytic structure (do they max extrema, self intersections, etc).
- Looking at low dimensional shapes and understanding the equations that could model them.
- Understanding in general how the features of an equation give rise to geometric and analytic structures that we may want to analyze.
I think that computer systems are decent for (1) since students can explore different equations and see how the geometry of their solutions change. I think they're much less helpful for (2) and virtually worthless for (3).
For (2) and (3), you have to understand the theory of why, say, both $z = x^2 - y^2$ and $z = xy$ give rise to saddles points. If you can manipulate the equation, draw the level curves and trace curves, compute the Jacobean and really understand how all those structures explicitly give rise to the behavior of the solutions to the equation, then you'll be able to start forming fundamental connections between equations and geometry. Computer visualizations are useful to check your work and to get an idea of the visual vocabulary, but at the end of the day the connections made rely on memorization, much like the quadratic zoo most text books have.
So the hard, but important, skill is doing geometry without a computer. You can spend class time showing students the easy crutch that frankly they can (and will) all figure out on their own, or you can spend your class time helping them through the actual hard work. Computer visualization systems and solvers are useful, and should be discussed in class, but the more you rely on them, the students will come to understand that background skill is worthless, when in fact it's just difficult.
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With the advent of the era of Wolfram Alpha.
A great deal is to calculate the curvature of a planar curve. The formula is $$\frac{x'y''-x''y'}{\sqrt{(x'^2+y'^2)^3}}$$ and a crash problem is calculate that for the projective coordinates of a circle of radius $R$ which is $$x=\frac{R(1-t^2)}{t^2+1}$$ $$y=\frac{2R}{t^2+1}.$$ Only very well concentrated minds can do the calculation without software aid, but with Wolfram Alpha every body can see how this calculation should be performed to be elegant.
The WA command for an specific, say $R=12$, is:
x=(12)(1-t^2)/(t^2+1); y=(24)t/(t^2+1); (D[x,t]*D[D[y,t],t]-D[y,t]*D[D[x,t],t])/(D[x,t]^2+D[y,t]^2)^(3/2)
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$\begingroup$ I agree with you that using Wolfram Alpha is easier than doing a problem by hand, but could you say a little bit more about how this teaches students about planer curves? $\endgroup$ Commented Aug 13, 2018 at 15:48
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$\begingroup$ One must push that they do by hand the first times but eventually they should learn assisted calculations or plots with a device at hand... Here two examples about plot in the afore mentioned free software: a plane curve wolframalpha.com/input/… and a spatial one wolframalpha.com/input/… and almost from them, lots of experimentations. This fashion, I believe, we will be encouraging to have "initiativessness" of any. $\endgroup$– janmarqzCommented Aug 13, 2018 at 20:27
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There is a great YouTube series on this topic from the Berkley graduate William Flannery called "Mathematical Modeling and Computational Calculus I" and "Mathematical Modeling and Computational Calculus II". He uses MATLAB to simulate different physics problems that involve calculus. The multivariable calculus comes more in the 2nd series when he starts working with the partial differential equations. He has also two books that support these courses and can be purchased on Amazon (Vol. 1 and Vol. 2). Finally, he has a website called Berkeley Science where you can find his other materials.
