## 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
$g$
.