It is good to have a spring break and to enjoy snow through
a window of my comfortable room. This morning I examined
an old French textbook "Electrostatique, Courants Continus,
Magnetisme" by P. Fleury and J.P. Mathieu. It was published
by Editions Eyrolles, Paris, in 1960. I was looking for a
confirmation that the "misconception of inversibilty" is not
limited to USA.
In that university textbook the V=B*Q matrix is introduced in
the context of the principle of superposition. Then the authors
say: "SOLVING THESE EQUATIONS WITH RESPECT TO
CHARGES, one can see that, inversely, charges are linear
functions of potentials." This is followed by the Q=C*V matrix.
It is interesting that diagonal elements of C are called "capacitances
of individual conductors". The non-diagonal elements are called
coefficients of influence. No special name is given to Bij.
It would be interesting to trace the origin of the misconception
exposed by John. A good history of science project, I suppose.
It would be useful if people who have access to old books could
share with us earlier examples of the misconception. It is true
that linear relations exist between charges and potentials in both
ways (when Q are independent variables, and when V are
independent variables). What is not true is that Cij can be
calculated from Bij, or vice versa, by the matrix inversion
method. The misconception is in the "SOLVING THESE
EQUATIONS" part of the description. Everything else is
usually OK, as far as I can tell.
By the way, referring to what John calls the "gauge invariance",
the above French authors wrote: "It can be shown that the
coefficient of influence of conductor i on conductor j is equal
to the coefficient of influence of conductor j on conductor i."
This is not as obvious as one may think at first. Consider a
very small and a very large objects influencing each other.
Ludwik Kowalski