Euler's Formula

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• Feb 13th 2013, 08:07 AM
lytwynk
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.
• Feb 13th 2013, 08:47 AM
emakarov
Re: Euler's Formula
Differentiate both sides of $f(tx,ty)=t^nf(x,t)$ 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 $e^{ix} = \cos x + i\sin x$ or $V - E + F = 2$ where V, E and F are respectively vertices, edges and faces of a convex polyhedron.
• Feb 13th 2013, 08:52 AM
johng
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

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

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

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

Evaluate at t = 1, QED.
• Feb 13th 2013, 09:15 AM
lytwynk
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.
• Feb 13th 2013, 09:19 AM
emakarov
Re: Euler's Formula
Quote:

Originally Posted by lytwynk
So the question is asking to start with the original definition of Euler's theorem?

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