I'd tend to define a mathematician as anyone who does mathematics--as removed from elementary arithmetic. Perhaps math enthusiasts aren't professional mathematicians but they are still mathematicians.
Though, that leads to tricky spots with other words: is anyone who ponders philosophy a philosopher?
At the end of the day, Leibniz notation is just that---notation. If you can get the correct intuition from it, great; if not, don't read more into it than is actually there.
@Ted: I don't know why but the problem in your book that asks for a parametrization of the circle by the y-intercept of the line going between a point on the circle and (-1,0) gave me a lot of trouble, but I cracked it.
if you write x=y=2, though, you can't regard it as a dummy variable. it's got a specific value.
there's probably some good analogy in programming, e.g. a variable that's defined in the course of some operation versus one that's assigned explicitly.
There's really three things one can do in that case. One can regard everything in there as a function of $x$, and therefore differentiate w/r/t x. Or one can regard it as a function of x, and differentiate w/r/t y.
or one can regard 3y+2 as a function of x and y. in that case you can take either derivative, but these will now be partial derivatives.
I just kept hunting too hard for an equation besides the one provided by similarity of triangles...and then realized $x^2 + y^2 = 1$ was sitting in front of me.
sorry i haven't been doing math and stuff, I've been trying to set up this personal website of mine with this "new" (from 1998) laptop my sister gave me
It has no value, because it could be either, tilted "downwards" or "upwards". A workaround for that would be "infty=-infty" for that case alone, but rather useless.
