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

exterior derivative of integral | 1.56 | 0.4 | 7773 | 66 | 31 |

exterior | 0.56 | 0.6 | 9134 | 36 | 8 |

derivative | 1.08 | 0.4 | 4370 | 44 | 10 |

of | 0.44 | 0.1 | 2714 | 74 | 2 |

integral | 0.65 | 0.5 | 5398 | 69 | 8 |

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 .

Then one sees that dω corresponds to a p + 1 exterior form. By the way, a natural and simple definition of tangent vector on a smooth manifold is given in the same book in (5.5.1). The exterior derivative is an intrinsic way of talking about the gradient of a function.

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 .

The derivative of an integral of a function is the function itself. But this is always true only in the case of indefinite integrals. The derivative of a definite integral of a function is the function itself only when the lower limit of the integral is a constant and the upper limit is the variable with respect to which we are differentiating.