# gradient theorem

If $u=u(x,y,z)$ is continuously differentiable function in a simply connected domain (http://planetmath.org/Domain2) $D$ of ${\mathbb{R}}^{3}$ and $P=({x}_{0},{y}_{0},{z}_{0})$ and $Q=({x}_{1},{y}_{1},{z}_{1})$ lie in this domain, then

${\int}_{P}^{Q}}\nabla u\cdot \overrightarrow{ds}=u({x}_{1},{y}_{1},{z}_{1})-u({x}_{0},{y}_{0},{z}_{0})$ | (1) |

where the line integral^{} of the left hand side is taken along an arbitrary path in $D$.

The equation (1) is illustrated by the fact that

$$\nabla u\cdot \overrightarrow{ds}=\frac{\partial u}{\partial x}dx+\frac{\partial u}{\partial y}dy+\frac{\partial u}{\partial z}dz$$ |

is the total differential^{} of $u$, and thus

$${\int}_{P}^{Q}\nabla u\cdot \overrightarrow{ds}={\int}_{P}^{Q}\mathit{d}u.$$ |

Title | gradient theorem |
---|---|

Canonical name | GradientTheorem |

Date of creation | 2013-03-22 19:11:16 |

Last modified on | 2013-03-22 19:11:16 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 26B12 |

Synonym | fundamental theorem of line integrals |

Related topic | LaminarField |

Related topic | Gradient |