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.

Differentiate both sides of 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 or where V, E and F are respectively vertices, edges and faces of a convex polyhedron.

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



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



Evaluate at t = 1, QED.

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.

Quote:

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Originally Posted by lytwynk

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

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No, you differentiate both sides of the equation that is the main part of the definition of a homogeneous function, namely, .