# Euler’s theorem on homogeneous functions

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

Let $f\mathit{}\mathrm{(}{x}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{x}_{k}\mathrm{)}$ be a smooth homogeneous function of degree $n$. That is,

$$f(t{x}_{1},\mathrm{\dots},t{x}_{k})={t}^{n}f({x}_{1},\mathrm{\dots},{x}_{k}).$$ | (*) |

Then the following identity holds

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

###### 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 $t$:

$${x}_{1}\frac{\partial f}{\partial {x}_{1}}(t{x}_{1},\mathrm{\dots},t{x}_{k})+\mathrm{\cdots}+{x}_{k}\frac{\partial f}{\partial {x}_{k}}(t{x}_{1},\mathrm{\dots},t{x}_{k})=n{t}^{n-1}f({x}_{1},\mathrm{\dots},{x}_{k}).$$ |

To obtain the result of the theorem, it suffices to set $t=1$ in the previous formula^{}.
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Sometimes the differential operator ${x}_{1}{\displaystyle \frac{\partial}{\partial {x}_{1}}}+\mathrm{\cdots}+{x}_{k}{\displaystyle \frac{\partial}{\partial {x}_{k}}}$ 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^{}.

Title | Euler’s theorem on homogeneous functions |
---|---|

Canonical name | EulersTheoremOnHomogeneousFunctions |

Date of creation | 2013-03-22 15:18:58 |

Last modified on | 2013-03-22 15:18:58 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 10 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 26B12 |

Classification | msc 26A06 |

Classification | msc 15-00 |

Defines | Euler operator |