Re: [Phys-L] notation for partial derivatives

[John Denker <jsd@av8n.com>]

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.