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Home > Exterior derivative

to differential forms of higher degree. It is important in the theory of integration on manifolds, and is the differential used to define de Rham and Alexander-Spanier cohomology. Its current form was invented by Élie Cartan.

1 Definition

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

For a k-form ω = f_I dx_I over Rⁿ, the definition is as follows:

For general k-forms Σ_I f_I dx_I (where the multi-index I runs over all ordered subsets of {1, ..., n} of cardinality k), we just extend linearly. Note that if above then (see wedge product).

2 Properties

Exterior differentiation satisfies three important properties:

• linearity

• the wedge product rule (see antiderivation)

• and d² = 0, a formula encoding the equality of mixed partial derivatives, so that always

It can be shown that exterior derivative is uniquely determined by these properties and its agreement with the differential on 0-forms (functions).

The kernelAlgebra In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. The definition of kernel takes various forms in various conte of d consists of the closed forms, and the imageThe image of an element x isin X under the function f ''X rarr Y denoted by f ''x , is the unique y isin Y that is associated with x''. The image of a subset A&sub X under f is the subset of Y defined by f A) f x) | x in A} Notice that the range of f is t of the exact forms (cf. exact differentials).

3 Invariant formula

Given a -form and arbitrary smooth vector fields we have

where denotes Lie bracketA lie bracket can refer to: Lie algebra Lie derivative. and

In particular, for 1-forms we have:

4 Connection with vector calculus

The following correspondence reveals about a dozen formulas from vector calculusMultivariate calculus Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics. We con as merely special cases of the above three rules of exterior differentiation.

4.1 GradientMultivariate calculus In vector calculus, gradient is a vector-valued operator that acts on a scalar field. The gradient of a scalar field is a vector field called gradient field which shows its rate and direction of change. For example, consider a room.

For a 0-form, that is a smooth function f: Rⁿ→R, we have

Therefore

where denotes gradient of and is scalar product.

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Topics: Exterior Derivative Differential Vector Product Form Forms...

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Exterior derivativeIn mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. It is important in the theory of integration on manifolds, and is the differential use Exterior algebraIn mathematics, the exterior algebra (also known as the Grassmann algebra of a given vector space V is a certain unital associative algebra which contains V as a subspace. It is denoted by Λ V or Λ• V and its multiplication, known as the wed Exterior Gateway ProtocolThe Exterior Gateway Protocol EGP is a routing protocol for the Internet originally specified in 1982 by Eric C. Rosen of Bolt, Beranek and Newman. It was first described in and formally specified in ( 1984). EGP had limited scalability and was eventually Exterior (topology)Topology General topology In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which do not meet S''. The exterior of S is denoted by ext ''S or S e''.