Anyway, a student e-mailed me a question today. One problem on the homework was to find equations in cylindrical and spherical coordinates for x2 + y2 + z2 + 2z = 0. In cylindrical coordinates, (where x2 + y2 = r2 this becomes r2 + z2 + 2z = 0. (My aesthetic sense is that this is perhaps better written as r2 + (z+1)2 = 1, because then it's immediately obvious that there are in fact solutions to the equation.) She asked if it was acceptable to leave it in one of these forms, or if it needed to be converted to, say, something with z in terms of r and θ (which would involve square roots).
I responded as follows:
In rectangular coordinates it might be preferable to solve for z in terms of x and y if it's possible to do so without making things too ugly, since we have a tendency to think of surfaces given in Cartesian coordinates as a "graph" of a function of x and y. The same thing is sort of true in spherical coordinates, in that ρ is often seen as a function of θ and φ (This is like polar coordinates, where r is usually a function of θ.) In cylindrical coordinates, though, none of the three coordinates really seem to be "dependent" or "independent".
Of course, all these rules can be violated; for example, one would never write the equation of a unit sphere centered at the origin as
z = ± (1-x2-y2)1/2
unless it were to graph it on a system that can't handle implicitly defined surfaces. The guiding principle should be that you want the simplest equation possible, i. e. the one which takes the least writing.
"Takes the least writing" is admittedly a bit sloppy here; how does one define it? The amount of "writing" any mathematical expression takes depends on one's notation. For example, subscripts or superscripts take up a lot of keystrokes in HTML, not so many in TeX (which is of course optimized for mathematical writing), and if one is writing by hand writing a subscript is just one character. (Indeed, one could even argue that subscripts and superscripts require less writing than full-size characters in handwriting -- they take less ink! But I shudder at the thought of taking this to its logical conclusion.)
Another principle I might invoke is "symmetry" -- another reason the equation about for a sphere looks wrong is because it fails to capture the fact that x, y, and z all behave "the same way". The form x2 + y2 + z2, on the other hand, doesn't have this problem. But I think "symmetry" might not be so useful a concept when trying to teach students; the ones who will understand this rather vague idea are the same ones who would probably just give the equation in a "more symmetric" form even if I didn't tell them to.
And why do cylindrical coordinates not seem to have a "dependent" and an "independent" variable, whereas three-dimensional Cartesian and spherical coordinates do? (I admit that this might just be my intuition, and I don't know if this agrees with anyone else's intuition.)
3 comments:
I think we are all just basically conditioned to think that Cartesian coordinates (at least in 2D and 3D) have a natural independent and dependent variable, even though that's not the case, mainly because of our experiences in learning coordinate systems. The vast majority of our exposure to coordinate systems is in the context of plotting graphs of functions, where we assign the independent variable to the "x" and the dependent variable to the "y". But that doesn't HAVE to be the way we assign variables, and there's no good reason why we should do it that way other than convention.
Anybody who's studied coordinate geometry for any length of time knows that x and y don't necessarily have to be related by a dependence, otherwise we'd never have lemniscates or elliptic curves. But (American, at least) students get algebra and functions first long before they get that kind of geometry, if indeed they get it at all.
I think there's a fairly intuitive (to me) "dependence" in cylindrical coordinates; at least as much, if not more, as between (x,y) and z in Cartesian coordinates.
If I wanted to express one cylindrical coordinate of a surface as a function of the other two, the "natural" choice to me would be r(z, θ). In this formulation, the constant function r(z, θ) = 1 defines the unit cylinder, and r(z, θ)^2 = 1 - z^2 defines the unit sphere.
The way I visualize it is as an open cylinder whose surface is perturbed as r(z, θ); this decomposition lends itself to integration in cylindrical coordinates nicely.
Jacob,
that seems reasonable... but why not consider, say, z(r, θ), where, for example, z(r, θ) = r is a cone?
I haven't thought about this from the point of view of integration. I probably should before my students get to integrating!
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