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# Euler’s theorem on homogeneous functions

###### Theorem 1 (Euler).

Let $f(x_{1},\ldots,x_{k})$ be a smooth homogeneous function of degree $n$. That is,

$f(tx_{1},\ldots,tx_{k})=t^{n}f(x_{1},\ldots,x_{k}).\tag{*}$ |

Then the following identity holds

$x_{1}\frac{\partial f}{\partial x_{1}}+\cdots+x_{k}\frac{\partial f}{\partial x% _{k}}=nf.$ |

###### Proof.

By homogeneity, the relation^{} (1) holds for all $t$. Taking the t-derivative of both sides, we establish that the following identity holds for all $t$:

$x_{1}\frac{\partial f}{\partial x_{1}}(tx_{1},\ldots,tx_{k})+\cdots+x_{k}\frac% {\partial f}{\partial x_{k}}(tx_{1},\ldots,tx_{k})=nt^{{n-1}}f(x_{1},\ldots,x_% {k}).$ |

To obtain the result of the theorem, it suffices to set $t=1$ in the previous formula^{}.
∎

Sometimes the differential operator $\displaystyle{x_{1}\frac{\partial}{\partial x_{1}}+\cdots+x_{k}\frac{\partial}%
{\partial x_{k}}}$ is called the *Euler operator*. An equivalent^{} way to state the theorem is to say that homogeneous functions are eigenfunctions of the Euler operator, with the degree of homogeneity as the eigenvalue^{}.

Defines:

Euler operator

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

## Mathematics Subject Classification

26B12*no label found*26A06

*no label found*15-00

*no label found*

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Incorrect Sequence by troworld ✓

contains own proof by mps ✓

Contains own proof by rm50 ✓

Contains own proof by rm50 ✓

contains own proof by mps ✓

Contains own proof by rm50 ✓

Contains own proof by rm50 ✓