[Phys-L] geometric interpretation of partial derivatives
From: jsd at AV8N.COM (John Denker)
Date: Mon Dec 20 12:34:05 2004
Hi --
Executive summary: Partial derivatives have many important uses in math
and science. We shall see that a partial derivative is not much more or
less than a particular sort of directional derivative. The only trick is
to have a reliable way of specifying directions ... so most of this note
is concerned with formalizing the idea of direction. This results in a
nice geometric way of visualizing the meaning of âpartial derivativeâ.
Partial derivatives are particularly confusing in non-Cartesian
coordinate systems, such as are commonly encountered in thermodynamics.
Drawing such a picture isn't hard; the important thing is to realize
that such a picture must exist. It must exist because partial
differentiation is a geometrically well-founded operation. It works
even in situations (such as thermo) where you have no dot product,
and therefore no notion of angle or distance.
Also there is a section
file:///home/jsd/physics/partial-derivative.htm#sec-vis
that discusses how to visualize directions in terms of vectors and
differential forms. This BTW answers some questions that came up about
a year ago, when people were asking about how the wedge product between
pointy vectors was related to the wedge product between one-forms.