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

exterior derivative of a 1 form | 1.63 | 0.5 | 7585 | 36 | 31 |

exterior | 0.58 | 0.4 | 2343 | 21 | 8 |

derivative | 0.32 | 0.5 | 3289 | 59 | 10 |

of | 1.78 | 0.5 | 2303 | 86 | 2 |

a | 1.32 | 0.2 | 7444 | 86 | 1 |

1 | 0.47 | 0.9 | 7112 | 81 | 1 |

form | 0.84 | 0.8 | 391 | 14 | 4 |

The exterior derivative is defined to be the unique ℝ -linear mapping from k -forms to (k + 1) -forms satisfying the following properties: df is the differential of f , for 0 -forms ( smooth functions) f . d(df ) = 0 for any 0 -form (smooth function) f .

Proposition 1 (Exterior Derivative of a 1-Form). For any smooth 1-form ω and smooth vector ﬁelds X and Y , (1) dω(X,Y)=X(ω(Y))−Y(ω(X))−ω([X,Y]). Proof. Since any smooth 1-form can be expressed locally as a sum of terms of the form udv for smooth functions u and v, it suﬃces to consider that case. Suppose ω = udv, and X, Y are smooth vector ﬁelds.

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 .

The above depicts the exterior derivative of a 1-form d φ ( v, w), which is the sum of φ along the boundary of the completed parallelogram defined by v and w. So if in the diagram ε = 1, we have d φ ( v, w) = ( 2 − 1) − ( 0 − 0) + 3 = 4.