Editing Differential form (section)

=== The exterior derivative ===
In addition to the exterior product, there is also the [[exterior derivative]] operator {{math|''d''}}.  The exterior derivative of a differential form is a generalization of the [[differential of a function]], in the sense that the exterior derivative of {{math|1=''f'' ∈ ''C''{{sup|∞}}(''M'') = Ω{{sup|0}}(''M'')}} is exactly the differential of {{mvar|f}}. When generalized to higher forms, if {{math|1=''ω'' = ''f'' ''dx''{{i sup|''I''}}}} is a simple {{mvar|k}}-form, then its exterior derivative {{math|''dω''}} is a {{math|(''k'' + 1)}}-form defined by taking the differential of the coefficient functions:
<math display="block">d\omega = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \, dx^i \wedge dx^I.</math>
with extension to general {{mvar|k}}-forms through linearity: if {{nowrap|<math display="inline">\tau = \sum_{I \in \mathcal{J}_{k,n}} a_I \, dx^I \in \Omega^k(M)</math>,}} then its exterior derivative is
<math display="block">d\tau = \sum_{I \in \mathcal{J}_{k,n}}\left(\sum_{j=1}^n \frac{\partial a_I}{\partial x^j} \, dx^j\right)\wedge dx^I \in \Omega^{k+1}(M)</math>

In {{math|'''R'''<sup>3</sup>}}, with the [[Hodge star operator]], the exterior derivative corresponds to [[gradient]], [[Curl (mathematics)|curl]], and [[divergence]], although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution. 

The exterior derivative itself applies in an arbitrary finite number of dimensions, and is a flexible and powerful tool with wide application in [[differential geometry]], [[differential topology]], and many areas in physics. Of note, although the above definition of the exterior derivative was defined with respect to local coordinates, it can be defined in an entirely coordinate-free manner, as an [[Derivation (differential algebra)|antiderivation]] of degree 1 on the [[exterior algebra]] of differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate on [[manifold]]s. It also allows for a natural generalization of the [[fundamental theorem of calculus]], called the (generalized) [[Stokes' theorem]], which is a central result in the theory of integration on manifolds.