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Necessary and Sufficient Conditions for a Function of Two Other Functions to be Equal to Zero

Theorem
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
\[g\]
.

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