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## Homework Statement

Let ##u## and ##v## be differentiable functions of ##x,~y## and ##z##. Show that a necessary and sufficient condition that ##u## and ##v## are functionally related by the equation ##F(u,v)=0## is that ##\vec \nabla u \times \vec \nabla v= \vec 0##

## Homework Equations

(Not applicable)

## The Attempt at a Solution

##\vec \nabla u## and ##\vec \nabla u## are the normal vectors to the constant ##u##-surface and the constant ##v##-surface respectively. As, ##\vec \nabla u \times \vec \nabla v= \vec 0##, i.e, ##\vec \nabla u## and ##\vec \nabla v## are in the same (or opposite) direction for a particular value of ##(x, y, z)##, a constant ##u##-surface also represents a constant ##v##-surface. Therefore, for a particular value of ##u##, there exists a corresponding value of ##v##. So, we can conclude that ##u## and ##v## are functionally related.

But, how can I prove it mathematically?