On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The resulting calculus, known as exterior calculus, allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.
If a differential -form is thought of as measuring the flux through an infinitesimal -parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a -parallelotope at each point.
Definition
The exterior derivative of a differential form of degree (also differential -form, or just -form for brevity here) is a differential form of degree .
If is a smooth function (a -form), then the exterior derivative of is the differential of . That is, is the unique -form such that for every smooth vector field , , where is the directional derivative of in the direction of .
The exterior product of differential forms (denoted with the same symbol ) is defined as their pointwise exterior product.
There are a variety of equivalent definitions of the exterior derivative of a general -form.
In terms of axioms
The exterior derivative is defined to be the unique -linear mapping from -forms to -forms that has the following properties:
is the differential of for a -form . for a -form . where is a -form. That is to say, is an antiderivation of degree on the exterior algebra of differential forms (see the graded product rule).
The second defining property holds in more generality: for any -form ; more succinctly, . The third defining property implies as a special case that if is a function and is a -form, then because a function is a -form, and scalar multiplication and the exterior product are equivalent when one of the arguments is a scalar.
In terms of local coordinates
Alternatively, one can work entirely in a local coordinate system . The coordinate differentials form a basis of the space of one-forms, each associated with a coordinate. Given a multi-index with for (and denoting with ), the exterior derivative of a (simple) -form
over is defined as
(using the Einstein summation convention). The definition of the exterior derivative is extended linearly to a general -form
where each of the components of the multi-index run over all the values in . Note that whenever equals one of the components of the multi-index then (see Exterior product).
The definition of the exterior derivative in local coordinates follows from the preceding definition in terms of axioms. Indeed, with the -form as defined above,
Here, we have interpreted as a -form, and then applied the properties of the exterior derivative.
This result extends directly to the general -form as
In particular, for a -form , the components of in local coordinates are
Caution: There are two conventions regarding the meaning of . Most current authors have the convention that
while in older text like Kobayashi and Nomizu or Helgason
In terms of invariant formula
Alternatively, an explicit formula can be given for the exterior derivative of a -form , when paired with arbitrary smooth vector fields :
where denotes the Lie bracket and a hat denotes the omission of that element:
In particular, when is a -form we have that .
Note: With the conventions of e.g., Kobayashi–Nomizu and Helgason the formula differs by a factor of :
Examples
Example 1. Consider over a -form basis for a scalar field . The exterior derivative is:
The last formula, where summation starts at , follows easily from the properties of the exterior product. Namely, .
Example 2. Let be a -form defined over . By applying the above formula to each term (consider and ) we have the following sum,
Stokes' theorem on manifolds
If is a compact smooth orientable -dimensional manifold with boundary, and is an -form on , then the generalized form of Stokes' theorem states that:
Intuitively, if one thinks of as being divided into infinitesimal regions, and one adds the flux through the boundaries of all the regions, the interior boundaries all cancel out, leaving the total flux through the boundary of .
Further properties
Closed and exact forms
A -form is called closed if ; closed forms are the kernel of . is called exact if for some -form ; exact forms are the image of . Because , every exact form is closed. The Poincaré lemma states that in a contractible region, the converse is true.
