Sorry to return to this after such a gap -I've been away out of contact
with the email. If everybody else is happy with the explanation offered by
Jannie & Johann, then I'll fall in with the majority. But note Athel's
recent posting - it's not clear that we are getting a sufficiently
widespread debate of these issues. On this particular point - total v
partial derivatives - I'm about 60% convinced.
In response to my query, Jannie & Johann wrote:
"However, if y is not a function of x, then dy/dx = 0 and we find
that the total derivative ***reduces*** to the partial derivative:
dz/dx = cz/cx. ..............(3)"
Now I don't disagree with this, but it seems to me that we we're in danger
of having this the wrong way round. My point - implicitly shown by the
derivation of equation 3 -was that it is the total derivative that we are
actually trying to arrive at - and the fact that it is a partial derivative
under certain particular circumstances doesn't necessarily justify
describing it as such in general.
Later on they say:
"Nothing stops one from describing the
steady-state flux as a function of rate functions, J =
J(v1,.....,vn), from which it follows that
dJ = (cJ/cv1)dv1 + .... + (cJ/cvn)dvn
This is quite acceptable from a mathematical point of view. Eq. 1
remains valid even is x and y are in turn functions of, say, u and
v. One can then use the chain rule to expand the equation for dz -
see next).
In turn, each rate function is a function of the parameters, so
that each
dvi = (cvi/cp1)dp1 + ..... + (cvi/cpn)dpn
which leads to
dJ = (cJ/cv1)[ (cv1/cp1)dp1 + ..... + (cv1/cpn)dpn ]
+ .... +
(cJ/cvn)[ (cvn/cp1)dp1 + ..... + (cvn/cpn)dpn ]"
It is eaxctly this point that worries me. When is it valid to make the
change of variables implied by (cJ/cv1)(cv1/cp1)dp1 ? In general, partial
derivatives need not be defined in such a way that this is universally
valid. There is no problem with (cJ/cv1)(dv1/dp1)dp1 because there is only
one set of restrictions applying to the term - those relevant to the first
partial derivative. If you have the product of two partial derivatives,
then you have to be sure that the restrictions applying to each are not
mutually exclusive, and that the combination makes sense.
One of Christoph Giersch's JTB papers had a case where this issue was
important and far from trivial.
Let me give a simple example where I think the total/partial distinction
matters - the derivation of the Heinrich & Rapoprt equation for a flux
control coefficient in terms of the elasticities and the concentration
control coefficients. Thus we are considering the flux at a step j, and
how this is influenced by a change in the rate at a step i.
vj = f(Vj, S1....Sn)
Then
dvj/dvi = dvj/dvi + (cvj/cS1)(dS1/dvi) + ... (cvj/cSn)(dSn/dvi)
=>
Control coeff of i on flux j = Sum(k=1..n) (elasticity j wrt Sk)(Control
coeff of Sk wrt i).
When i=j, there is a 1 on the LHS.
To my mind, this is transparent and obviously correct in terms of a
combination of partial and total derivatives, and would be obscure as
combinations of different types of partial derivatives. (For example, how
do you explain that (dS1/dvi) can be understood as (CS1/cvi) the control
coefficient, but not (CS1/cvi) the reciprocal of the elasticity?)
But if I'm really the only one still worried by this........
David
