Consider a relation between x, y and z of the type
F ( u, v) = 0 (12.22)
where u and v are known functions of x, y and z;
and F is an arbitrary function of u and v. Also, z is a
function of x and y.
Differentiating (12.22) by chain rule, with
respect to x and y we obtain
0
FuuFvv
pp
uxzvxz
∂∂∂∂∂∂
⎛⎞⎛⎞
+++=
⎜⎟⎜⎟
⎝⎠⎝⎠
∂∂∂∂∂∂
(12.23)
0
FuuFvv
qq
uyzvyz
⎛⎞⎛⎞
∂∂∂∂∂∂
+++=
⎜⎟⎜⎟
∂
⎝
∂∂
⎠
∂
⎝
∂∂
⎠
(12.24)
Eliminating F between (12.23) and ...
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