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# Re: [Phys-L] notation for partial derivatives

• 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

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.