Question

state Euler's theorem for homogeneous function
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Answer 1

Answer:

There is a theorem, usually credited to Euler, concerning homogenous functions that we might be making use of.

A homogenous function of degree n of the variables x, y, z is a function in which all terms are of degree n. For example, the function f(x, y, z)=Ax3+By3+Cz3+Dxy2+Exz2+Gyx2+Hzx2+Izy2+Jxyz is a homogenous function of x, y, z, in which all terms are of degree three.

The reader will find it easy to evaluate the partial derivatives ∂f∂x, ∂f∂x, ∂f∂x and equally easy (if slightly tedious) to evaluate the expression x∂f∂x+y∂f∂y+z∂f∂z . Tedious or not, I do urge the reader to do it. You should find that the answer is 3Ax3+3By3+3Cz3+3Dxy2+3Exz2+3Fyz2+3Gyx2+3Hzx2+3Izy2+3Jxyz.

In other words, x∂f∂x+y∂f∂y+z∂f∂z=3f . If you do the same thing with a homogenous function of degree 2, you will find that x∂f∂x+y∂f∂y+z∂f∂z=2f . And if you do it with a homogenous function of degree 1, such as Ax+By+Cz , you will find that x∂f∂x+y∂f∂y+z∂f∂z=f . In general, for a homogenous function of x, y, z... of degree n, it is always the case that

x∂f∂x+y∂f∂y+z∂f∂z+...=nf.(2.6.1)

This is Euler's theorem for homogenous functions.