Re: [Phys-l] heat +- impulse

[John Denker <jsd@av8n.com>]

On 11/04/2007 12:13 AM, LaMontagne, Bob wrote:

> What comes to mind is heat as a tranfer variable - it only has
> existence while energy is lost in one region and is gained in
> another. Work is also a similar quantity. It only exists when a force
> acts to move an object, but results in a change in a state variable,
> energy.

Let's see if I understand what is meant by "transfer variable".

1) A discrete "change in energy" is obviously not a function of state;
it is a function of two states:
        ΔE = E(B) - E(A)                [1]
referring to the two states (A) and (B).

I hereby assume that's the point.  (If that's not the point,
please explain ... and please ignore the rest of this note.)



2) We agree that point (1) is a valid point.  

Alas, it seems to be something of a negative point;  it tells
us what we can't do with ΔE, but it fails to tell us what we 
/can/ do.

In my experience, it is tremendously useful (practically and
pedagogically) to pass to the limit where point (A) and point 
(B) are very close together.  Then, with suitable normalization, 
ΔE becomes dE, where dE is a vector, in particular a gradient 
vector in state-space.

The nice thing is that dE is a function of state!  Therefore
it is a huge improvement over ΔE which was a function of two 
states.

At the next level of detail: the dE vector is a one-form (not 
a pointy vector) and as such it is best visualized in terms of 
contour lines, as shown at
  http://www.av8n.com/physics/thermo-forms.htm#fig-bump
and
  http://www.av8n.com/physics/thermo-forms.htm#fig-dV
and
  http://www.av8n.com/physics/thermo-forms.htm#fig-dT


In any case, this allows us to interpret an equation such as
     dE = p dV - T dS                   [2]
as a _vector_ equation.

Note that the RHS of equation [2] has a work-related term and
a heat-related term.  

Also note that each of these terms is a function of state!
 -- T is a scalar function of state
 -- S is a scalar function of state
 -- dS is a vector function of state
 -- T dS is a vector function of state 


3) To repeat, I'm not saying point (1) is wrong.  I'm just
suggesting that it doesn't lead anywhere super-useful.  There's 
not much constructive you can do with it.  

Therefore I'm suggesting a shift in emphasis.  Rather than
emphasizing "transfer variable", IMHO it is more helpful to
emphasize "vector".  Once you realize that T dS is a vector,
there's a lot you can do with it.

BTW note that the ancient roots of the word "vector" are not
too different in meaning from "transfer".  Vector means,
literally, "carrier" (as in convection, and as in mosquitoes
which are a vector for malaria).  So it's really quite a
small step to go from "transfer variable" to vector.