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Re: [Phys-L] notation for partial derivatives



On 06/21/2014 07:10 AM, Diego Saravia wrote:

dE dV componentes of the "diferencial of state vector"

I assume the word "differential" denotes a generic
/infinitesimal/ change ... which is not the approach
I would recommend.

By way of contrast, the expression ΔV denotes a
/difference/, a finite difference ... not a
differential, not an infinitesimal. As such, it is
not a function of state. In fact, it is a function
of two states, A and B:
ΔV = V(A) - V(B)

If V is a scalar, then ΔV is also a scalar.

Returning from ΔV to dV, the derivative dV is a
function of state. It is defined *at* a single
point in state-space (and its infinitesimal
neighborhood). Interestingly, dV is not a scalar.
It must be considered a vector, for reasons we
now discuss.

In many cases, given a formula involving ΔV there
is a "similar" formula involving dV ... but don't
be fooled. The interpretation of dV is quite
different from the interpretation of ΔV.

V scalar function of state
ΔV scalar not a function of state
dV vector function of state

The idea that in a multi-variable space dV has to be
considered a vector comes as a surprise to students.
The same idea applies in D=1, but nobody ever notices,
because in D=1 there is not much difference between
scalars and vectors. There is a one-to-one mapping
from scalars to vectors on the number line.

The derivative does not even live in the same space
as the thing being differentiated; instead it lives
in the /tangent space/. Here's a picture:
http://standards.sedris.org/18026/text/ISOIEC_18026E_SRF/image022.jpg
The same idea applies to functions of a single
variable, but introductory calculus courses sweep
it under the rug.

To repeat: It is unhelpful to think of dV as a
generic "differential" i.e. a generic infinitesimal
change. To make sense of it, you need to specify
the /direction/ in which things are changing. This
fact is central to the questions that started this
thread.

Because derivatives are linear, there is an easy
and elegant way to specify the direction. The
trick is to think of dV as a vector. Thereafter,
to find the amount of change, all we need to do
is form the contraction between dV and some vector
specifying the direction in which things are
changing.

In spaces where there is a metric, i.e. where
there is a well-defined notion of angle and
distance, you can replace the word "contraction"
with "dot product" ... but remarkably, the whole
dV-is-a-vector approach works even when there
is no metric (e.g. thermodynamics) or where the
metric is unknown or unpleasant (e.g. general
relativity).

If the previous sentence doesn't mean anything
to you, don't worry about it.

There is tremendous power in this approach. For
starters, an equation such as

dE = -P dV - T dS [1]

is a vector equation. It does not specify a
direction of change; it expresses an idea
that is valid for each and every direction.
As such, it contains a myriad of special cases,
including changes in the direction of constant
V (where dV=0) and alternatively changes in
the direction of constant S (where dS=0) and
even changes in the direction of constant E
(where dE=0).

For hundreds of years, mathematicians tried to
find a workable interpretation of dV in terms
of infinitesimal scalars ... tried and failed.

For the next level of detail, see
http://www.av8n.com/physics/differential-forms.htm
http://www.av8n.com/physics/thermo-forms.htm