# fundamental lemma of calculus of variations

The idea in the calculus of variations^{} is to study
stationary points of functionals.
To derive a differential equation^{} for such stationary
points, the following theorem is needed, and hence
named thereafter. It is also used in distribution theory
to recover traditional calculus from distributional calculus.

###### Theorem 1.

Suppose $f\mathrm{:}U\mathrm{\to}\mathrm{C}$ is a locally integrable function on an open subset $U\mathrm{\subset}{\mathrm{R}}^{n}$, and suppose that

$${\int}_{U}f\varphi \mathit{d}x=0$$ |

for all smooth functions with compact support $\varphi \mathrm{\in}{C}_{\mathrm{0}}^{\mathrm{\infty}}\mathit{}\mathrm{(}U\mathrm{)}$. Then $f\mathrm{=}\mathrm{0}$ almost everywhere.

By linearity of the integral, it is easy to see that one only needs to
prove the claim for real $f$. If $f$ is continuous^{}, this can be seen
by purely geometrical arguments. A full proof
based on the Lebesgue differentiation theorem is given
in [1]. Another proof is given in [2].

## References

Title | fundamental lemma of calculus of variations^{} |
---|---|

Canonical name | FundamentalLemmaOfCalculusOfVariations |

Date of creation | 2013-03-22 15:02:04 |

Last modified on | 2013-03-22 15:02:04 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 8 |

Author | matte (1858) |

Entry type | Theorem |

Classification | msc 28B15 |

Synonym | fundamental theorem of the calculus of variations |

Related topic | CalculusOfVariations |