# Green’s theorem

Green’s theorem provides a connection between path integrals over a well-connected region in the plane and the area of the region bounded in the plane. Given a closed path $P$ bounding a region $R$ with area $A$, and a vector-valued function^{} $\overrightarrow{F}=(f(x,y),g(x,y))$ over the plane,

$${\oint}_{P}\overrightarrow{F}\cdot \mathit{d}\overrightarrow{x}=\int {\int}_{R}[{g}_{1}(x,y)-{f}_{2}(x,y)]\mathit{d}A$$ |

where ${a}_{n}$ is the derivative of $a$ with respect to the $n$th variable.

## Corollary:

The closed path integral over a gradient of a function with continuous^{} partial derivatives^{} is always zero. Thus, gradients are conservative vector fields. The smooth function^{} is called the potential of the vector field.

## Proof:

The corollary states that

$${\oint}_{P}{\overrightarrow{\nabla}}_{h}\cdot \mathit{d}\overrightarrow{x}=0$$ |

We can easily prove this using Green’s theorem.

$${\oint}_{P}{\overrightarrow{\nabla}}_{h}\cdot \mathit{d}\overrightarrow{x}=\int {\int}_{R}[{g}_{1}(x,y)-{f}_{2}(x,y)]\mathit{d}A$$ |

But since this is a gradient…

$$\int {\int}_{R}[{g}_{1}(x,y)-{f}_{2}(x,y)]\mathit{d}A=\int {\int}_{R}[{h}_{21}(x,y)-{h}_{12}(x,y)]\mathit{d}A$$ |

Since ${h}_{12}={h}_{21}$ for any function with continuous partials, the corollary is proven.

Title | Green’s theorem |
---|---|

Canonical name | GreensTheorem |

Date of creation | 2013-03-22 12:15:55 |

Last modified on | 2013-03-22 12:15:55 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 10 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 26B20 |

Related topic | GaussGreenTheorem |

Related topic | ClassicalStokesTheorem |