Re: [Phys-L] f'(t) / g'(t) question

[John Denker <jsd@av8n.com>]

In this forum there is a tradition of pursuing various tangents,
ramifications, and additional viewpoints.  There is considerable
value in this.  It has been known in the pedagogical psychology
literature for well over 100 years that the more ways you have
of thinking about something, the more useful it is, and the more
likely it is to be remembered (James, 1898).


On 04/08/2016 07:57 AM, Carl Mungan wrote:

> http://www.usna.edu/Users/physics/mungan/_files/documents/Scholarship/PartialDerivativeVanDerWaals.pdf

Yes, the technique used there is intimately connected to the
question that started this thread.  Indeed there are several
layers of connections.

> What’s interesting about this example is that the method of solution
> I present here, which I would argue is relatively straightforward and
> logical for a physicist, drives mathematician’s crazy. They insist on
> much more convoluted (call it rigorous if you don’t want to appear
> partisan) approaches.

  A) those mathematicians are making an important point, but
  B) they're not being very constructive about it.

People have tried for 350 years to come up with a consistent theory
of "differentials" aka "infinitesimals".  Tried and failed.

Constructive suggestion:  In the aforementioned PDF, if you replace
one word, it becomes vastly more rigorous, without becoming convoluted.
Replace "differentials" with "exterior derivatives".  If you reinterpret
all the equations using this idea, they all work just fine.

In particular, it must be emphasized that if you want to make sense
of thermodynamics, then things like dT, dS, dV, et cetera must be 
considered vectors.  More specifically, they are gradient vectors.
Even more specifically, they are one-forms (as opposed to pointy 
vectors).

It stands to reason that you cannot talk about an infinitesimal change
in V without specifying the /direction/ of the change.

Note the contrast:
    V is a scalar and a function of state.
   ΔV = V2-V1 is a scalar but not a function of state.
        It is at best a function of two states (state 1 and state 2).
   dV is a function of state but not a scalar.
        It has direction as well as magnitude.
        If you didn't keep track of direction it wouldn't be a
        function of state.

Most of the people who study thermodynamics already know about vectors,
so there's virtually no cost in doing things this way.

For more on this, see
  https://www.av8n.com/physics/thermo-forms.htm
  https://www.av8n.com/physics/non-grady.htm

About 85% of the equations in a typical thermo book can be reinterpreted
in this way.  The other 15% are just wrong and should be thrown out.  For 
example, the equation dQ = T dS is just wrong.  There is no state-function
Q that satisfies such an equation (except in trivial cases).  Schroeder
rightly calls such things a crime against the laws of mathematics.

When key equations in the textbook are provably wrong, we should not
be surprised that people find the topic incomprehensible.  Indeed, if
you find dQ = T dS incomprehensible, I say congratulations, it shows
you're paying attention.

The good news is that you can throw out the offending equations and
still do thermodynamics just fine.  For example, if you mean T dS
just write T dS;  don't pretend it is equal to dQ.  Forget about Q
and dQ.