2.6: Euler's Theorem for Homogeneous Functions

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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 \( f(x,~y,~z) = Ax^3 +By^3+Cz^3+Dxy^2+Exz^2+Gyx^2+Hzx^2+Izy^2+Jxyz\) 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 \( \frac{\partial f}{\partial x},~ \frac{\partial f}{\partial x},~ \frac{\partial f}{\partial x}\) and equally easy (if slightly tedious) to evaluate the expression \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z}\). Tedious or not, I do urge the reader to do it. You should find that the answer is \( 3Ax^3 +3By^3+3Cz^3 + 3Dxy^2+3Exz^2+3Fyz^2+3Gyx^2+3Hzx^2+3Izy^2+3Jxyz.\)

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

\[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} + z \frac{\partial f}{\partial z} + ... = nf.\]

This is Euler's theorem for homogenous functions.