Editing Exterior algebra (section)

=== Differential geometry ===
The exterior algebra has notable applications in [[differential geometry]], where it is used to define [[differential form]]s.<ref>{{cite book |first=A.T. |last=James |chapter=On the Wedge Product |title=Studies in Econometrics, Time Series, and Multivariate Statistics |editor-first=Samuel |editor-last=Karlin |editor2-first=Takeshi |editor2-last=Amemiya |editor3-first=Leo A. |editor3-last=Goodman |publisher=Academic Press |year=1983 |isbn=0-12-398750-4 |pages=455–464 |chapter-url=https://books.google.com/books?id=-hDjBQAAQBAJ&pg=PA455 }}</ref> Differential forms are mathematical objects that evaluate the length of vectors, areas of parallelograms, and volumes of [[Parallelepiped#Parallelotope|higher-dimensional bodies]], so they can be [[integral|integrated]] over curves, surfaces and higher dimensional [[manifold]]s in a way that generalizes the [[line integral]]s and [[surface integral]]s from calculus.  A [[differential form]] at a point of a [[differentiable manifold]] is an alternating multilinear form on the [[tangent space]] at the point.  Equivalently, a differential form of degree {{math|''k''}} is a [[linear functional]] on the {{math|''k''}}th exterior power of the tangent space.  As a consequence, the exterior product of multilinear forms defines a natural exterior product for differential forms.  Differential forms play a major role in diverse areas of differential geometry.

An [[Differential (mathematics)#Differentials as germs of functions|alternate approach]] defines differential forms in terms of [[Germ (mathematics)|germs of functions]].

In particular, the [[exterior derivative]] gives the exterior algebra of differential forms on a manifold the structure of a [[differential graded algebra]].  The exterior derivative commutes with [[pullback (differential geometry)|pullback]] along smooth mappings between manifolds, and it is therefore a [[natural transformation|natural]] [[differential operator]].  The exterior algebra of differential forms, equipped with the exterior derivative, is a [[cochain complex]] whose cohomology is called the [[de Rham cohomology]] of the underlying manifold and plays a vital role in the [[algebraic topology]] of differentiable manifolds.