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*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 â€œpartial derivativeâ€.

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.

**Follow-Ups**:**[Phys-L] Re: geometric interpretation of partial derivatives***From:*jsd at AV8N.COM (John Denker)

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