Necessary and sufficient conditions for functions
\[f(x,y), g(x,y)\]
to be functionally related by the equation \[F(f(x,y), g(x,y))=0\]
is \[\nabla f(x,y) \times \nabla g(x,y)=\mathbf{0}\]
Proof
Assume
\[F(f(x,y), g(x,y))=0\]
and differentiate with respect to \[x\]
to give \[\frac{\partial F}{\partial f} \frac{\partial f}{\partial x}+\frac{\partial F}{\partial g} \frac{\partial g}{\partial x} =0\]
and with respect to \[y\]
to give \[\frac{\partial F}{\partial f} \frac{\partial f}{\partial y}+\frac{\partial F}{\partial g} \frac{\partial g}{\partial y} =0\]
.\[\frac{\partial F}{\partial f} \neq 0\]
and \[\frac{\partial F}{\partial g} \neq 0\]
. these equations are only consistent if the determinant of the matrix of coefficents is zero, so that \[ \left| \begin{array}{cc} \frac{\partial f}{\partial x} & \frac{\partial g}{\partial x} \\ \frac{\partial f}{\partial y} & \frac{\partial g}{\partial y} \end{array} \right| = \frac{\partial (f,g)}{\partial(x,y)}=0\]
.But
\[\nabla f(x,y) \times \nabla g(x,y)=\left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial f}{\partial x} & \frac{\partial g}{\partial x} &0 \\ \frac{\partial f}{\partial y} & \frac{\partial g}{\partial y} &0 \end{array} \right| = \mathbf{k} \frac{\partial (f,g)}{\partial(x,y)}=0\]
.If
\[\nabla f(x,y) \times \nabla g(x,y)=\mathbf{0}\]
then \[f, g\]
have the same level curves. The equations \[h=f(x,y), k=g(x,y)\]
define a transformation from the \[xy\]
plane to the \[hk\]
plane. The transformation is degenerate because along each level curve, \[h, k\]
are constant and the transformation maps to a single point in the \[hk\]
plane.Let
\[(x_0 , y_0 )\]
be a point in the domain of \[f,g\]
. Since \[\nabla f \neq 0\]
, \[f\]
must either increase or decrease when moving from this point in a direction normal to the level curve. The same argument holds for \[g\]
.