Re: [Phys-L] notation for partial derivatives

[John Denker <jsd@av8n.com>]

On 06/20/2014 01:05 PM, Carl Mungan wrote:
> I agree with the idea that one ought to specify what variable is
> being held in a partial derivative if there is any possibility of
> confusion.

Indeed!  Agreed!

The "conventional" notation for partial derivatives 
is a horror show.

[ several relevant examples snipped ]

> These are just warmup examples. You can probably come up with much
> more gnarly examples. For example, maybe I want to specify that one
> is to substitute in some value such as x0 for x at some point along
> the way (say I have 3 variables x,y,x).

Let me throw a little more fuel on the fire:

What do we mean when we write z(x,y)?  Do we mean that 
z(x,y) is a function of x and y, for all x and y?  Or 
do we mean that z(x,y) is the value we get when we apply
the function z to the particular values x and y?

And yet some more fuel:

What do you do with (say) a Lagrangian that depends
on some variables and on some derivatives?  What
does it mean to differentiate something with respect
to the derivative of x?

========

I don't have time to put the details into email right
now, but suffice it to say there is tremendous merit
in the approach taken in
  Sussman and Wisdom
  _Structure and Interpretation of Classical Mechanics_

See especially their appendix on notation:
  http://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-Z-H-79.html
especially notation for derivatives:
  http://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-Z-H-79.html#%_sec_Temp_452
and partial derivatives:
  http://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-Z-H-79.html#%_sec_Temp_453

Quick summary:  They talk about taking the derivative
of a function with respect to its Nth argument-position.

Remark:  Even if you don't 100% love their notation, it
serves as an existence proof:  A systematic, reliable,
non-nonsense notation is possible.

Tangential remark:  The standards of rigor and formality
in /computer science/ far exceed the standards of normal
mathematics or physics.  When you want to tell a computer
what you mean, it forces you to be very systematic and
explicit.  Sussman and Wisdom discovered that most of
the calculations in classical physics books could be
converted to computer language ... but others could 
not, because they were simply not correct.  They 
"looked" correct, but they weren't.

S & W coined the term "proof by pun" referring to alleged
proofs that throw around words that would have been OK
in another context, but not in the context where they
are actually being used.

==================

To answer a question that wasn't directly asked, but
is guaranteed to come up before long in this context:

Oftentimes it pays to think of /functions of state/
that depend on a highly abstract /state/.  This is
important (and possibly familiar) in the context of
thermodynamics.  Things like E, F, G, and H exist as 
functions of state.  It is unnecessary and unhelpful
to think of E as a function of S and V to the exclusion
of other things it might be a function of.  Instead
think of E(state).  Maybe the state is known as a
function of S and V, and/or maybe as a function of
various other things, but in any case it is the
/state/ that matters.  It is a point in a high-
dimensional abstract space.

This is a big win, because it allows us to write things
like
     dE = -P dV - T dS             [1]
which is true no matter what (if anything) you consider
"the" independent variables.

This doesn't solve all of the world's problems, but
it helpfully hides some of them, at least temporarily.
In particular, when we define P in this context, we 
have to define it as -∂E/∂V _at constant S_ with due 
regard for the constant S.

Equation [1] has a delightful pictorial interpretation 
in terms of vectors in an abstract high-dimensional 
space.  Let's be clear: dE, dV, dS, and similar things
must be considered vectors.

The pattern repeats itself:  Most of the standard
results of thermodynamics can easily be converted to
this way of doing things.  The few that cannot were
not correct to begin with!