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*From*: John Denker <jsd@av8n.com>*Date*: Sat, 21 Jun 2014 12:09:35 -0700

On 06/21/2014 10:26 AM, Diego Saravia wrote:

Are you meaning dV? or (dV,dU)

I assume dV is a component, ok could be a one dimmension vector,

but the tangent space in the all mighty S(U,V) could be (dU,dV)

That's definitely not what I was talking about.

I say dV is a vector.

Hypothetically, you could construct some higher-grade

object of which the dV vector is a "component",

but I don't see why you would bother. I don't

see any physical significance in that, or any

operational advantage.

ok, you could be talking of a vector in gradient terms in another

space, for example

in a P,T representation depaV/depaP and depaV/depaT is also, a vector

with components

in dP and dT coordinates.

That's also not what I was talking about. dP and

dT are not coordinates.

=========

I say again:

At any point in state space, dP is a vector.

Considering all of state space collectively,

dP is a vector field.

Given that the pressure (P) is a scalar field, dP

is the /gradient/ of P. dP is equivalent to ∇P.

More generally, dX is ∇∧X, but if X happens

to be a scalar field then dX, ∇∧X, and ∇X are

the same thing.

As an example, d(height) is well represented by

the contour lines on a topographic map.

This topic is called differential topology. dX

is called the exterior derivative of X. It is not

something I just made up on a whim. There are

fat books on the subject. If you have a copy of

Misner,Thorn,Wheeler _Gravitation_ lying around

(which you should; it's a really good book) it

offers a reasonable introduction to the subject

along with insight as to the physical significance.

Also this document

http://viper.princeton.edu/~ssgubser/courses/Ph106a01/handouts/forms.pdf

is a quick overview (not a tutorial) covering the

bits that are most relevant to physics. Or (gasp)

you could look at the references I cited last time:

http://www.av8n.com/physics/differential-forms.htm

http://www.av8n.com/physics/thermo-forms.htm

and (especially relevant to this thread)

http://www.av8n.com/physics/partial-derivative.htm

The field of differential topology as a whole is

more elaborate than most students want or need,

but the basic ideas are simple and powerful. Some

of the ideas are used throughout physics already.

This includes the following distinction:

row vectors are different from column vectors

bras are different from kets (in Dirac notation)

et cetera.

Differential topology was developed mostly in the

early 20th century. It will probably take another

100 years before it gets used (or even mentioned)

in introductory textbooks.

**Follow-Ups**:**Re: [Phys-L] notation for partial derivatives***From:*Diego Saravia <dsa@unsa.edu.ar>

**References**:**[Phys-L] notation for partial derivatives***From:*Carl Mungan <mungan@usna.edu>

**Re: [Phys-L] notation for partial derivatives***From:*John Denker <jsd@av8n.com>

**Re: [Phys-L] notation for partial derivatives***From:*Diego Saravia <dsa@unsa.edu.ar>

**Re: [Phys-L] notation for partial derivatives***From:*John Denker <jsd@av8n.com>

**Re: [Phys-L] notation for partial derivatives***From:*Diego Saravia <dsa@unsa.edu.ar>

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