Exterior derivative Guide, Meaning , Facts, Information and Description

In 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 used to define de Rham and Alexander-Spanier cohomology. Its current form was invented by Élie Cartan.

Table of contents

1 Definition 2 Properties 3 Invariant formula 4 Connection with vector calculus 4.1 Gradient 4.2 Curl 4.3 Divergence 5 Examples 6 See also

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).

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 kernel of d consists of the closed forms, and the image of the exact forms (cf. exact differentials).

Invariant formula

Given a -form and arbitrary smooth vector fields we have

where denotes Lie bracket and

In particular, for 1-forms we have:

Connection with vector calculus

The following correspondence reveals about a dozen formulas from vector calculus as merely special cases of the above three rules of exterior differentiation.

Gradient

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

Therefore

where denotes gradient of and is scalar product.

Curl

For a 1-form on R^3,

which restricted to the three-dimensional case is

Therefore, for vector field we have

the where denotes curl of ,

is the vector product and  is scalar product.

Divergence

For a 2-form

For three dimensions, with we get

where V is a vector field defiend by

Examples

For a 1-form on R² we have

which is exactly the 2-form being integrated in Green's theorem.

See also

• Exterior covariant derivative
• Green's theorem
• Stoke's theorem

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