Keyword | CPC | PCC | Volume | Score | Length of keyword |
---|---|---|---|---|---|

exterior derivative examples | 0.22 | 0.8 | 2204 | 53 | 28 |

exterior | 0.1 | 0.2 | 8002 | 93 | 8 |

derivative | 0.78 | 0.6 | 5636 | 37 | 10 |

examples | 0.72 | 0.5 | 2592 | 39 | 8 |

Keyword | CPC | PCC | Volume | Score |
---|---|---|---|---|

exterior derivative examples | 0.84 | 0.6 | 5780 | 72 |

The exterior derivative of a function is the one-form. (1) written in a coordinate chart . Thinking of a function as a zero-form, the exterior derivative extends linearly to all differential k -forms using the formula. (2) when is a -form and where is the wedge product . The exterior derivative of a -form is a -form.

The exterior derivative of this 0 -form is the 1 -form df . When an inner product ⟨·,·⟩ is defined, the gradient ∇f of a function f is defined as the unique vector in V such that its inner product with any element of V is the directional derivative of f along the vector, that is such that

If f is a smooth function (a 0 -form), then the exterior derivative of f is the differential of f . That is, df is the unique 1 -form such that for every smooth vector field X, df (X) = dX f , where dX f is the directional derivative of f in the direction of X .

Thinking of a function as a zero-form, the exterior derivative extends linearly to all differential k -forms using the formula when is a -form and where is the wedge product . The exterior derivative of a -form is a -form. For example, for a differential k -form where is the permutation tensor . It is always the case that .