From Wikipedia, the free encyclopedia
Distributions on spaces of differential forms
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 functional on the space of compactly supported differential k-forms, on a smooth manifold M. Currents formally behave like Schwartz distributions on a space of differential forms, but in a geometric setting, they can represent integration over a submanifold, generalizing the Dirac delta function, or more generally even directional derivatives of delta functions (multipoles) spread out along subsets of M.
Let Ω c m ( M ) {\displaystyle \Omega _{c}^{m}(M)} denote the space of smooth m-forms with compact support on a smooth manifold M . {\displaystyle M.} A current is a linear functional on Ω c m ( M ) {\displaystyle \Omega _{c}^{m}(M)} which is continuous in the sense of distributions. Thus a linear functional T : Ω c m ( M ) → R {\displaystyle T:\Omega _{c}^{m}(M)\to \mathbb {R} } is an m-dimensional current if it is continuous in the following sense: If a sequence ω k {\displaystyle \omega _{k}} of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when k {\displaystyle k} tends to infinity, then T ( ω k ) {\displaystyle T(\omega _{k})} tends to 0.
The space D m ( M ) {\displaystyle {\mathcal {D}}_{m}(M)} of m-dimensional currents on M {\displaystyle M} is a real vector space with operations defined by ( T + S ) ( ω ) := T ( ω ) + S ( ω ) , ( λ T ) ( ω ) := λ T ( ω ) . {\displaystyle (T+S)(\omega ):=T(\omega )+S(\omega ),\qquad (\lambda T)(\omega ):=\lambda T(\omega ).}
Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current T ∈ D m ( M ) {\displaystyle T\in {\mathcal {D}}_{m}(M)} as the complement of the biggest open set U ⊂ M {\displaystyle U\subset M} such that T ( ω ) = 0 {\displaystyle T(\omega )=0} whenever ω ∈ Ω c m ( U ) {\displaystyle \omega \in \Omega _{c}^{m}(U)}
The linear subspace of D m ( M ) {\displaystyle {\mathcal {D}}_{m}(M)} consisting of currents with support (in the sense above) that is a compact subset of M {\displaystyle M} is denoted E m ( M ) . {\displaystyle {\mathcal {E}}_{m}(M).}
Homological theory[edit]Integration over a compact rectifiable oriented submanifold M (with boundary) of dimension m defines an m-current, denoted by [ [ M ] ] {\displaystyle [[M]]} : [ [ M ] ] ( ω ) = ∫ M ω . {\displaystyle [[M]](\omega )=\int _{M}\omega .}
If the boundary ∂M of M is rectifiable, then it too defines a current by integration, and by virtue of Stokes' theorem one has: [ [ ∂ M ] ] ( ω ) = ∫ ∂ M ω = ∫ M d ω = [ [ M ] ] ( d ω ) . {\displaystyle [[\partial M]](\omega )=\int _{\partial M}\omega =\int _{M}d\omega =[[M]](d\omega ).}
This relates the exterior derivative d with the boundary operator ∂ on the homology of M.
In view of this formula we can define a boundary operator on arbitrary currents ∂ : D m + 1 → D m {\displaystyle \partial :{\mathcal {D}}_{m+1}\to {\mathcal {D}}_{m}} via duality with the exterior derivative by ( ∂ T ) ( ω ) := T ( d ω ) {\displaystyle (\partial T)(\omega ):=T(d\omega )} for all compactly supported m-forms ω . {\displaystyle \omega .}
Certain subclasses of currents which are closed under ∂ {\displaystyle \partial } can be used instead of all currents to create a homology theory, which can satisfy the Eilenberg–Steenrod axioms in certain cases. A classical example is the subclass of integral currents on Lipschitz neighborhood retracts.
Topology and norms[edit]The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence. A sequence T k {\displaystyle T_{k}} of currents, converges to a current T {\displaystyle T} if T k ( ω ) → T ( ω ) , ∀ ω . {\displaystyle T_{k}(\omega )\to T(\omega ),\qquad \forall \omega .}
It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If ω {\displaystyle \omega } is an m-form, then define its comass by ‖ ω ‖ := sup { | ⟨ ω , ξ ⟩ | : ξ is a unit, simple, m -vector } . {\displaystyle \|\omega \|:=\sup\{\left|\langle \omega ,\xi \rangle \right|:\xi {\mbox{ is a unit, simple, }}m{\mbox{-vector}}\}.}
So if ω {\displaystyle \omega } is a simple m-form, then its mass norm is the usual L∞-norm of its coefficient. The mass of a current T {\displaystyle T} is then defined as M ( T ) := sup { T ( ω ) : sup x | | ω ( x ) | | ≤ 1 } . {\displaystyle \mathbf {M} (T):=\sup\{T(\omega ):\sup _{x}|\vert \omega (x)|\vert \leq 1\}.}
The mass of a current represents the weighted area of the generalized surface. A current such that M(T) < ∞ is representable by integration of a regular Borel measure by a version of the Riesz representation theorem. This is the starting point of homological integration.
An intermediate norm is Whitney's flat norm, defined by F ( T ) := inf { M ( T − ∂ A ) + M ( A ) : A ∈ E m + 1 } . {\displaystyle \mathbf {F} (T):=\inf\{\mathbf {M} (T-\partial A)+\mathbf {M} (A):A\in {\mathcal {E}}_{m+1}\}.}
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.
Recall that Ω c 0 ( R n ) ≡ C c ∞ ( R n ) {\displaystyle \Omega _{c}^{0}(\mathbb {R} ^{n})\equiv C_{c}^{\infty }(\mathbb {R} ^{n})} so that the following defines a 0-current: T ( f ) = f ( 0 ) . {\displaystyle T(f)=f(0).}
In particular every signed regular measure μ {\displaystyle \mu } is a 0-current: T ( f ) = ∫ f ( x ) d μ ( x ) . {\displaystyle T(f)=\int f(x)\,d\mu (x).}
Let (x, y, z) be the coordinates in R 3 . {\displaystyle \mathbb {R} ^{3}.} Then the following defines a 2-current (one of many): T ( a d x ∧ d y + b d y ∧ d z + c d x ∧ d z ) := ∫ 0 1 ∫ 0 1 b ( x , y , 0 ) d x d y . {\displaystyle 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.}
This article incorporates material from Current on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
RetroSearch is an open source project built by @garambo | Open a GitHub Issue
Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo
HTML:
3.2
| Encoding:
UTF-8
| Version:
0.7.4