- #1
kostoglotov
- 234
- 6
Homework Statement
Show that if f is homogeneous of degree n, then
[tex] x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y) [/tex]
Hint: use the Chain Rule to diff. f(tx,ty) wrt t.
2. The attempt at a solution
I know that if f is homogeneous of degree n then [tex] t^nf(x,y) = f(tx,ty) [/tex]
But I'm really at a conceptual loss here.
I've tried the hint, I let f(tx, ty) = f(a,b) so a = tx and b = ty then
[tex] \frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}\frac{\partial a}{\partial t}+\frac{\partial f}{\partial b}\frac{\partial b}{\partial t} \\
\frac{\partial f}{\partial t} = \frac{\partial f}{\partial a}x + \frac{\partial f}{\partial b}y [/tex]
And I have tried looking at
[tex] \frac{d}{dt}t^nf(x,y) = nt^{n-1}f(x,y) [/tex]
But beyond this I just cannot see what I need to do.
Thanks.