Exterior derivative – Wikipedia

operation of differentiation in differential geometry

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899; it allows for a natural, metric-independent generalization of Stokes’ theorem, Gauss’s theorem, and Green’s theorem from vector calculus.

If a k-form is thought of as measuring the flux through an infinitesimal k-parallelotope, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k + 1)-parallelotope.


The exterior derivative of a differential form of degree k is a differential form of degree k + 1.

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

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