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Physics LibreTexts

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)=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 fx, fx, fx and equally easy (if slightly tedious) to evaluate the expression xfx+yfy+zfz. 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, xfx+yfy+zfz=3f. If you do the same thing with a homogenous function of degree 2, you will find that xfx+yfy+zfz=2f. And if you do it with a homogenous function of degree 1, such as Ax+By+Cz, you will find that xfx+yfy+zfz=f. In general, for a homogenous function of x, y, z... of degree n, it is always the case that

xfx+yfy+zfz+...=nf.

This is Euler's theorem for homogenous functions.


This page titled 2.6: Euler's Theorem for Homogeneous Functions is shared under a CC BY-NC license and was authored, remixed, and/or curated by Jeremy Tatum.

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