11.3 Formulas involving partial derivatives 579Teachers and textbooks providing instruction in more advanced mathe-
matics like to presume that their disciples know enough about functions
and partial derivatives to be able to write the equations (11.36) and
(11.37). We are not, at the present time, required to comprehend the
various reasons why these equations are important, but we can makean
observation. In case the first of the conditions
(11.38)ax ay Ozl
as as as ;
ax ay az
as as as I0 0,
ax ay az
a7 a7 a7a« a a7
ax ax ax
jaa as a7
jay ay ay
a« as a7
az az az3.6 0,is satisfied, the system (11.36) of equations can be solved to obtain
formulas expressing au/ax, au/ay, au/az in terms of au/aa, au/a3,
au/a7. Similarly, if the second condition in (11.38) is satisfied, then
the system (11.37) of equations can be solved to obtain formulas express-
ing au/aa, au/a(3, au/a7 in terms of au/ax, au/ay, au/az. The determi-
nants in (11.38) are called functional determinants or Jacobi determinants
or Jacobians, and those who are destined to encounter them later may
become thankful for this preliminary glimpse of them. Those accus-
tomed to use of matrices prefer to see the systems (11.36) and (11.37)
of equations written in the forms
au ax ay az
as as as as
au ax ay az
ea as as ap
an ax ay az
a7 a7 a7 a7au au as 00 a7
ax ax ax axax
au au as a, a7
ay ay ay ay ay
au au as afl a7
az az az azazau
as
au
as
au
ayThis permits use of the algebra of matrices to which an introduction was
given at the end of Section 2.5.
In the following problems, it is assumed that each of the given func-
tions has all of the derivatives and partial derivatives we want to use
and that these derivatives are continuous.
Problems 11.39
1 Assume that x, y, z are functions of a, 0, 7 such thatfi(x,Y.z,aA7) = 0 f2(x,Y,z,a,$,7) = 0 f3(x,Y,z,aA7) = 0.Write the equations obtained by taking partial derivatives with respect to a
and then find the condition under which these equations uniquely determine