# Exterior derivative

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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; it allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.

If a *k*-form is thought of as measuring the flux through an infinitesimal *k*-parallelepiped, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (*k* + 1)-parallelepiped.

## Definition

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

If *f* is a smooth function (a 0-form), then the exterior derivative of *f* is the differential of *f* . That is, d*f* is the unique 1-form such that for every smooth vector field *X*, d*f* (*X*) = d_{X} *f* , where d_{X} *f* is the directional derivative of *f* in the direction of *X*.

There are a variety of equivalent definitions of the exterior derivative of a general *k*-form.

### Axioms for the exterior derivative

The exterior derivative is defined to be the unique **R**-linear mapping from *k*-forms to (*k* + 1)-forms satisfying the following properties:

- d
*f*is the differential of*f*for smooth functions*f*. - d(d
*f*) = 0 for any smooth function*f*. - d(
*α*∧*β*) = d*α*∧*β*+ (−1)^{p}(*α*∧ d*β*) where Template:Mvar is a*p*-form. That is to say, d is an antiderivation of degree 1 on the exterior algebra of differential forms.

The second defining property holds in more generality: in fact, d(d*α*) = 0 for any *k*-form Template:Mvar; more succinctly, d^{2} = 0. The third defining property implies as a special case that if *f* is a function and Template:Mvar a *k*-form, then d(*fα*) = d(*f* ∧ *α*) = d*f* ∧ *α* + *f* ∧ d*α* because functions are 0-forms, and scalar multiplication and the exterior product are equivalent when one of the arguments is a scalar.

### Exterior derivative in local coordinates

Alternatively, one can work entirely in a local coordinate system (*x*^{1}, ..., *x ^{n}*). First, the coordinate differentials d

*x*

^{1}, ..., d

*x*form a basic set of one-forms within the coordinate chart. The formulas in this section rely on the Einstein summation convention. Given a multi-index

^{n}*I*= (

*i*

_{1}, ...,

*i*

_{k}) with 1 ≤

*i*≤

_{p}*n*for 1 ≤

*p*≤

*k*(and an abuse of notation d

*x*), the exterior derivative of a

^{I}*k*-form

over **R**^{n} is defined as

For a general *k*-form

where the components of the multi-index *I* run over all the values in {1, ..., *n*}, the definition of the exterior derivative is extended linearly. Note that whenever *i* is one of the components of the multi-index *I* then d*x ^{i}* ∧ d

*x*= 0 (see wedge product).

^{I}The definition of the exterior derivative in local coordinates follows from the preceding definition. Indeed, if *ω* = *f _{I}* d

*x*

^{i1}∧ ... ∧ d

*x*, then

^{ik}Here, we have interpreted *f _{I}* as a 0-form, and then applied the properties of the exterior derivative.

### Invariant formula

Alternatively, an explicit formula can be given for the exterior derivative of a *k*-form *ω*, when paired with *k* + 1 arbitrary smooth vector fields *V*_{0},*V*_{1}, ..., *V*_{k}*:*

where [*V _{i}*,

*V*] denotes the Lie bracket and a hat denotes the omission of that element:

_{j}In particular, for 1-forms we have: d*ω*(*X*, *Y*) = *Xω*(*Y*) − *Yω*(*X*) − *ω*([*X*, *Y*]), where *X* and *Y* are vector fields.

## Stokes' theorem on manifolds

If *M* is a compact smooth orientable *n*-dimensional manifold with boundary, and *ω* is an (*n* − 1)-form on *M*, then the generalized form of Stokes' theorem states that:

Intuitively, if one thinks of *M* 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 *M*.

## Examples

**Example 1.** Consider *σ* = *u* d*x*^{1} ∧ d*x*^{2} over a 1-form basis d*x*^{1}, ..., d*x ^{n}*. The exterior derivative is:

The last formula follows easily from the properties of the wedge product. Namely, d*x ^{i}* ∧ d

*x*= 0.

^{i}**Example 2.** Let *σ* = *u* d*x* + *v* d*y* be a 1-form defined over **R**^{2}. By applying the above formula to each term (consider *x*^{1} = *x* and *x*^{2} = *y*) we have the following sum,

## Further properties

### Closed and exact forms

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A *k*-form *ω* is called *closed* if d*ω* = 0; closed forms are the kernel of d. *ω* is called *exact* if *ω* = d*α* for some (*k* − 1)-form *α*; exact forms are the image of d. Because d^{2} = 0, every exact form is closed. The Poincaré lemma states that in a contractible region, the converse is true.

### de Rham cohomology

Because the exterior derivative d has the property that d^{2} = 0, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The *k*-th de Rham cohomology (group) is the vector space of closed *k*-forms modulo the exact *k*-forms; as noted in the previous section, the Poincaré lemma states that these vector spaces are trivial for a contractible region, for *k* > 0. For smooth manifolds, integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology over **R**. The theorem of de Rham shows that this map is actually an isomorphism, a far-reaching generalization of the Poincaré lemma. As suggested by the generalized Stokes' theorem, the exterior derivative is the "dual" of the boundary map on singular simplices.

### Naturality

The exterior derivative is natural in the technical sense: if *f* : *M* → *N* is a smooth map and Ω^{k} is the contravariant smooth functor that assigns to each manifold the space of *k*-forms on the manifold, then the following diagram commutes

so d(*f* ^{∗}*ω*) = *f* ^{∗}d*ω*, where *f* ^{∗} denotes the pullback of *f* . This follows from that *f* ^{∗}*ω*(·), by definition, is *ω*(*f*_{∗}(·)), *f*_{∗} being the pushforward of *f* . Thus d is a natural transformation from Ω^{k} to Ω^{k+1}.

## Exterior derivative in vector calculus

Most vector calculus operators are special cases of, or have close relationships to, the notion of exterior differentiation.

### Gradient

A smooth function *f* : **R**^{n} → **R** is a 0-form. The exterior derivative of this 0-form is the 1-form

That is, the form d*f* acts on any vector field *V* by outputting, at each point, the scalar product of *V* with the gradient ∇*f* of *f* .

The 1-form d*f* is a section of the cotangent bundle, that gives a local linear approximation to *f* in the cotangent space at each point.

### Divergence

A vector field *V* = (*v*_{1}, *v*_{2}, ... *v _{n}*) on

**R**

^{n}has a corresponding (

*n*− 1)-form

where denotes the omission of that element.

(For instance, when *n* = 3, in three-dimensional space, the 2-form *ω _{V}* is locally the scalar triple product with

*V*.) The integral of

*ω*over a hypersurface is the flux of

_{V}*V*over that hypersurface.

The exterior derivative of this (*n* − 1)-form is the *n*-form

### Curl

A vector field *V* on **R**^{n} also has a corresponding 1-form

Locally, *η _{V}* is the dot product with

*V*. The integral of

*η*along a path is the work done against −

_{V}*V*along that path.

When *n* = 3, in three-dimensional space, the exterior derivative of the 1-form *η _{V}* is the 2-form

### Invariant formulations of grad, curl, div, and Laplacian

On any Riemannian manifold, the standard vector calculus operators can be written in coordinate-free notation as follows:

where is the Hodge star operator and and are the musical isomorphisms.

## See also

- Exterior covariant derivative
- de Rham complex
- Discrete exterior calculus
- Green's theorem
- Lie derivative
- Stokes' theorem
- Fractal derivative

## References

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