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| In [[mathematics]], more particularly in [[functional analysis]], [[differential topology]], and [[geometric measure theory]], a '''k-current''' in the sense of [[Georges de Rham]] is a [[linear functional|functional]] on the space of [[compactly supported]] [[differential form|differential k-forms]], on a [[smooth manifold]] ''M''. Formally currents behave like [[Schwartz distribution]]s on a space of differential forms. In a geometric setting, they can represent integration over a submanifold, generalizing the [[Dirac delta function]], or more generally even [[directional derivative]]s of delta functions ([[multipole]]s) spread out along subsets of ''M''.
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| ==Definition==
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| Let <math>\Omega_c^m(M)</math> denote the space of smooth ''m''-[[differential form|forms]] with [[compact support]] on a [[smooth manifold]] <math>M</math>. A current is a [[linear functional]] on <math>\Omega_c^m(M)</math> which is continuous in the sense of [[distribution (mathematics)|distribution]]s. Thus a linear functional
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| :<math>T\colon \Omega_c^m(M)\to \mathbb{R}</math>
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| is an ''m''-current if it is [[continuous function|continuous]] in the following sense: If a sequence <math>\omega_k</math> of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when <math>k</math> tends to infinity, then <math>T(\omega_k)</math> tends to 0.
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| The space <math>\mathcal D_m(M)</math> of ''m''-dimensional currents on <math>M</math> is a [[real number|real]] [[vector space]] with operations defined by
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| :<math>(T+S)(\omega):= T(\omega)+S(\omega),\qquad (\lambda T)(\omega):=\lambda T(\omega).</math>
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| Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the '''support''' of a current <math>T \in \mathcal{D}_m(M)</math> as the complement of the biggest [[open set]] <math>U \subset M</math> such that
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| :<math>T(\omega) = 0 </math> whenever <math>\omega \in \Omega_c^m(U)</math> | |
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| The [[linear subspace]] of <math>\mathcal D_m(M)</math> consisting of currents with support (in the sense above) that is a compact subset of <math>M</math> is denoted <math>\mathcal E_m(M)</math>.
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| ==Homological theory==
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| [[Integral|Integration]] over a compact [[rectifiable set|rectifiable]] [[orientation (mathematics)|oriented]] submanifold ''M'' ([[manifold with boundary|with boundary]]) of dimension ''m'' defines an ''m''-current, denoted by <math>[[M]]</math>:
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| :<math>[[M]](\omega)=\int_M \omega.\,</math>
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| If the [[Boundary (topology)|boundary]] ∂''M'' of ''M'' is rectifiable, then it too defines a current by integration, and by virtue of [[Stokes' theorem]] one has:
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| :<math>[[\partial M]](\omega) = \int_{\partial M}\omega = \int_M d\omega = [[M]](d\omega).</math>
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| This relates the [[exterior derivative]] ''d'' with the [[boundary operator]] ∂ on the [[homology (mathematics)|homology]] of ''M''.
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| In view of this formula we can ''define'' a '''boundary operator''' on arbitrary currents
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| :<math>\partial\colon \mathcal D_{m+1} \to \mathcal D_m</math>
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| by
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| :<math>(\partial T)(\omega) := T(d\omega)\,</math>
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| for all compactly supported (''m''−1)-forms ω.
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| ==Topology and norms==
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| The space of currents is naturally endowed with the [[weak-* topology]], which will be further simply called ''weak convergence''. A [[sequence]] ''T''<sub>''k''</sub> of currents, [[limit point|converges]] to a current ''T'' if
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| :<math>T_k(\omega) \to T(\omega),\qquad \forall \omega.\,</math>
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| It is possible to define several [[norm (mathematics)|norms]] on subspaces of the space of all currents. One such norm is the ''mass norm''. If ω is an ''m''-form, then define its '''comass''' by
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| :<math> \|\omega\| := \sup\{|\langle \omega,\xi\rangle|\colon\xi \mbox{ is a unit, simple, }m\mbox{-vector}\}.</math>
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| So if ω is a [[tensor rank|simple]] ''m''-form, then its mass norm is the usual L<sup>∞</sup>-norm of its coefficient. The '''mass''' of a current ''T'' is then defined as
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| :<math>\mathbf M (T) := \sup\{ T(\omega)\colon \sup_x |\vert\omega(x)|\vert\le 1\}. </math>
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| The mass of a current represents the ''weighted area'' of the generalized surface. A current such that '''M'''(''T'') < ∞ is representable by integration over a suitably weighted rectifiable submanifold. This is the starting point of [[homological integration]].
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| An intermediate norm is Whitney's ''flat norm'', defined by
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| :<math> \mathbf F (T) := \inf \{\mathbf M(T - \partial A) + \mathbf M(A) \colon A\in\mathcal E_{m+1}\}.</math>
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| Two currents are close in the mass norm if they coincide away from a small part. On the other hand they are close in the flat norm if they coincide up to a small deformation.
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| ==Examples==
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| Recall that
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| :<math>\Omega_c^0(\mathbb{R}^n)\equiv C^\infty_c(\mathbb{R}^n)\,</math>
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| so that the following defines a 0-current:
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| :<math>T(f) = f(0).\,</math>
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| In particular every [[signed measure|signed]] [[regular measure|regular]] [[measure (mathematics)|measure]] <math>\mu</math> is a 0-current:
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| :<math>T(f) = \int f(x)\, d\mu(x).</math>
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| Let (''x'', ''y'', ''z'') be the coordinates in ℝ<sup>3</sup>. Then the following defines a 2-current (one of many):
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| :<math> T(a\,dx\wedge dy + b\,dy\wedge dz + c\,dx\wedge dz) = \int_0^1 \int_0^1 b(x,y,0)\, dx \, dy.</math>
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| ==See also==
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| *[[Georges de Rham]]
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| *[[Herbert Federer]]
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| *[[Differential geometry]]
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| *[[Varifold]]
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| ==References==
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| * {{citation
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| |first=G.
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| |last=de Rham
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| |author-link=Georges de Rham
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| |title=Variétés Différentiables
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| |series= Actualites Scientifiques et Industrielles
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| |volume= 1222
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| |publisher=Hermann
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| |place=Paris
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| |year=1973
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| |edition=3rd
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| |pages=X+198
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| |language= French
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| |mr=
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| |zbl=0284.58001
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| }}.
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| * {{citation
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| | last = Federer
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| | first = Herbert
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| | authorlink = Herbert Federer
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| | title = Geometric measure theory
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| | place= Berlin–Heidelberg–New York
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| | publisher = [[Springer-Verlag]]
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| | series = Die Grundlehren der mathematischen Wissenschaften
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| | volume = 153
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| | year = 1969
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| | pages = xiv+676
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| | isbn = 978-3-540-60656-7
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| | id=
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| | mr=0257325
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| | zbl= 0176.00801
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| }}.
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| * {{citation
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| |first=H.
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| |last=Whitney
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| |author-link=Hassler Whitney
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| |title=Geometric Integration Theory
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| |series=Princeton Mathematical Series
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| |volume=21
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| |publisher=[[Princeton University Press]] and [[Oxford University Press]]
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| |place=Princeton, NJ and London
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| |year=1957
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| |pages= XV+387
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| |mr=0087148
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| |zbl=0083.28204
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| }}.
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| * {{Citation
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| | last1=Lin
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| | first1=Fanghua
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| | last2=Yang
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| | first2= Xiaoping
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| | title=Geometric Measure Theory: An Introduction
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| | series= Advanced Mathematics (Beijing/Boston)
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| | volume=1
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| | place=Beijing/Boston
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| | pages=x+237
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| | publisher=Science Press/International Press
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| | isbn=978-1-57146-125-4
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| | year=2003
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| | mr=2030862
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| | zbl=1074.49011
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| }}
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| {{PlanetMath attribution|id=5980|title=Current}}
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| [[Category:Differential topology]]
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| [[Category:Generalized manifolds]]
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