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## Homework Statement

Let n be a positive integer. A function F is called honogeneous of degree n if it satisfies the equation F(tx,ty) = t

^{n}F(x,y) for all real t. Suppose f(x,y) has continuous second-order partial derivatives.

Show that if F is homogeneous of degree n, then

x*F_x + y*F_y = n*F(x,y), where F_x,F_y are the partial derivatives

with respect to x,y.

## Homework Equations

## The Attempt at a Solution

Suppose I let u=tx and v=ty. Then,

F_t = F_u * x + F_v * y,

which should then be equal to

n*t^(n-1) * F(x,y).

This, I think, almost looks like what I want to prove. Dividing by

t^(n-1) gives n*F(x,y) and (F_u * x + F_v * y)/(t^(n-1)), which I want

to rewrite as

x*F_x + y*F_y.

But I have no idea if/why this should be true. Am I thinking about this correctly, or have I done it the wrong way?