It can be shown that exterior derivative is uniquely determined by these properties and its agreement with the differential on 0forms (functions).
Special cases of exterior differentiation correspond to familiar differential operators of vector calculus along the same lines as the differential corresponds to the gradient. For example, in 3 dimensional Euclidean space, exterior derivative of a 1form corresponds to curl and exterior derivative of a 2form corresponds to divergence.
This correspondence reveals about a dozen formulas from vector calculus as merely special cases of the above three rules of exterior differentiation. The kernel of d consists of the closed forms, and the image of the exact forms (cf. exact differentials).
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