1. ## Euler's Formula

Hello, I have a mid term tomorrow and this question was on the practice mid term and our prof has not posted any solutions. I tried to catch him after class but there was a million other students crowding him. If there is anyone who could give me a hand with this I would appreciate it.

Show that if f is any homogeneous function of degree n, then it
satis es Euler's formula

xD1f(x; y) + yD2f(x; y) = nf(x; y):

Hint: Treat each side of the de ning equation as a function of 3
variables t; x and y, and use the chain rule to compute the partials
with respect to t. Then set t = 1.

2. ## Re: Euler's Formula

Differentiate both sides of $\displaystyle f(tx,ty)=t^nf(x,t)$ with respect to t using the chain rule, then set t = 1.

Wikipedia calls this fact Euler's homogeneous function theorem. Euler's formula usually refers to either $\displaystyle e^{ix} = \cos x + i\sin x$ or $\displaystyle V - E + F = 2$ where V, E and F are respectively vertices, edges and faces of a convex polyhedron.

3. ## Re: Euler's Formula

With appropriate restraints on the functions involved, the chain rule states:
Let u and v be functions of x, y and t, f a function of u and v and g(x,y,t) = f(u(x,y,t),v(x,y,t). Then

$\displaystyle g_t(x,y,t)=f_1(u(x,y,t),v(x,y,t))u_t(x,y,t)+f_2(u( x,y,t),v(x,y,t))v_t(x,y,z)$

So let f be homogenous; i.e. f(tx,ty)=t^nf(x,y). The partials w.r.t t give the equation:

$\displaystyle f_1(tx,ty)x+f_2(tx,ty)y=nt^{n-1}f(x,y)$

Evaluate at t = 1, QED.

4. ## Re: Euler's Formula

So the question is asking to start with the original definition of Euler's theorem? Then just differentiate wrt t? I understand that but I thought I had to differentiate the equation given wrt t then show that LS=RS.

5. ## Re: Euler's Formula

Originally Posted by lytwynk
No, you differentiate both sides of the equation that is the main part of the definition of a homogeneous function, namely, $\displaystyle f(tx,ty)=t^nf(x,t)$.