# closed differential forms on a simply connected domain

Let $D\subset {\mathbb{R}}^{2}$ be an open set and let $\omega $ be a differential form^{} defined on $D$.

###### Theorem 1

If $D$ is simply connected and $\omega $ is a closed differential form, then $\omega $ is an exact differential form.

The proof of this result is a consequence of the following useful lemmas.

###### Lemma 1

Let $\omega $ be a closed differential form
and suppose that ${\gamma}_{\mathrm{0}}$ and ${\gamma}_{\mathrm{1}}$ are two regular^{} homotopic curves in $D$ (with the same end points). Then

$${\int}_{{\gamma}_{0}}\omega ={\int}_{{\gamma}_{1}}\omega .$$ |

###### Lemma 2

Let $\omega $ be a continuous^{} differential form.
If given any two curves ${\gamma}_{\mathrm{0}}$, ${\gamma}_{\mathrm{1}}$ in $D$ with the same end-points,
it holds

$${\int}_{{\gamma}_{0}}\omega ={\int}_{{\gamma}_{1}}\omega ,$$ |

then $\omega $ is exact.

See the Poincaré Lemma for a generalization^{} of this result on $n$-dimensional manifolds.

Title | closed differential forms on a simply connected domain |
---|---|

Canonical name | ClosedDifferentialFormsOnASimplyConnectedDomain |

Date of creation | 2013-03-22 13:32:46 |

Last modified on | 2013-03-22 13:32:46 |

Owner | paolini (1187) |

Last modified by | paolini (1187) |

Numerical id | 14 |

Author | paolini (1187) |

Entry type | Theorem |

Classification | msc 53-00 |

Related topic | ClosedCurveTheorem |

Related topic | PoincareLemma |