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[Phys-L] geometric interpretation of partial derivatives



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 “partial derivative”.

Partial derivatives are particularly confusing in non-Cartesian
coordinate systems, such as are commonly encountered in thermodynamics.

I just wrote up my notes on the geometric interpretation of partial
derivatives, and put them at:
http://www.av8n.com/physics/partial-derivative.htm

Perhaps you will find some pedagogical value in the picture
http://www.av8n.com/physics/partial-derivative.htm#fig-partial-deriv

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.