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Calculus and Beyond Homework Help
Functional relation between u(x,y,z) and v(x,y,z)
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[QUOTE="arpon, post: 5443405, member: 526026"] [h2]Homework Statement [/h2] 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## [h2]Homework Equations[/h2] (Not applicable) [h2]The Attempt at a Solution[/h2] ##\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? [/QUOTE]
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Functional relation between u(x,y,z) and v(x,y,z)
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