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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 .What is the p + 1 exterior derivative of dω?
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.What is the exterior derivative of a smooth vector field?
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 .What is the derivative of an integral of a function?
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.