(^578) Partial derivatives
can tell us that J is a function of the independent variables x and y,
the "fixed variables" being displayed at the bottom of the vertical line.
The second symbol can tell us that the independent variables are x and
y and that the partial derivative with respect to x is to be evaluated
at the place for which x = y = 0. No sane person will bother with the
embellishments in a very long calculation in which au/ax and au/ay
must be written many times and their meanings are perfectly clear, but
the embellishments are available when they are needed.
Without introducing hordes of symbols for functions, we look briefly
at applications of the chain rule in situations in which u is a function of
x, y, z and, in addition, x, y, z are functions of a, i3, y. The assumption
following (11.31) applies here, and accordingly u is also a function of
a, $, y. While such agreements are usually made without explicit men-
tion, we solemnly proclaim that whenever the partial derivative with
respect to a Roman coordinate x or y or z (or a Greek coordinate a or
or y) appears, the thing being differentiated must be a function of those
coordinates and the other coordinates of the same nationality are the
"fixed variables." If we keep # and y so rigidly fixed that it is unneces-
sary to take this fact into account in our notation, then u is a function
of x, y, z and x, y, z are functions of a and the version of the chain rule in
Theorem 11.24 enables us to write
du au dx au dy au dz
da-8xda+ayda+azda
When we use partial derivative notation to convey the information that
and y are fixed, we obtain the first of the formulas
au au ax au ay au'az
as axaa+ayda+azaa
au au ax au ay au 19Z
(11.36)
a_=axa1+aya0+azas
au_au ax au ay au az
ay ax ay + ay ay + az ay'
and the next two are obtained by similar processes. If, as usually hap-
pens in applications, we confine attention to regions over which the equa-
tions giving x, y, z in terms of a, #,,y can be solved to givea, 0,,y in terms
of x, y, z then the same procedure gives the equationsau au as au ao au ay
axaaax+a1ax+ayax(11.37) au au as au ao au ay
ay asay+as ay+_
ay ay
au au as au aft au ay