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## Homework Statement

Let F(x,y) be a twice differentiable function such that

4 * F

_{x}

^{2}+ F

_{y}

^{2}= 0.

Set x = u

^{2}- v

^{2}and y = u*v. Show that

F

_{u}

^{2}+ F

_{v}

^{2}= 0

## The Attempt at a Solution

F

_{u}= F

_{x}*2u + F

_{y}*v

F

_{v}= F

_{x}*-2v + F

_{y}*u

F

_{u}

^{2}= 4v

^{2}F

_{x}

^{2}-4F

_{x}F

_{y}*u*v+F

_{y}

^{2}u

^{2}

F

_{v}

^{2}= 4u

^{2}F

_{x}

^{2}+4F

_{x}F

_{y}*u*v+F

_{y}

^{2}v

^{2}

Adding these two together gives

4F

_{x}

^{2}*(u

^{2}v

^{2}) + 4F

_{y}

^{2}*(u

^{2}v

^{2}),

Which from our first equation, can clearly equal zero if we factor out the (u

^{2}v

^{2}) and divide. Then, we see that, indeed, F

_{u}

^{2}+ F

_{v}

^{2}= 0.

Sound about right?