# Exterior derivative

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.

## Contents

## 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

**R**

^{n}, the definition is as follows:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d{\omega} = \sum_{i=1}^n \frac{\partial f_I}{\partial x_i} dx_i \wedge dx_I.}**

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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = I}**
above then **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dx_i \wedge dx_I = 0}**
(see wedge product).

## Properties

Exterior differentiation satisfies three important properties:

- the wedge product rule (see antiderivation)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(\omega \wedge \eta) = d\omega \wedge \eta+(-1)^{{\rm deg\,}\omega}(\omega \wedge d\eta)}**

- and
*d*^{2}= 0, a formula encoding the equality of mixed partial derivatives, so that always

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(d\omega)=0 \, \!}**

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 *k*-form *ω* and arbitrary smooth vector fields *V _{0},V_{1}, …, V_{k}* we have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\omega(V_0,V_1,...V_k)=\sum_i(-1)^i V_i\omega(V_0,...,\hat V_i,...,V_k)}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\sum_{i<j}(-1)^{i+j}\omega([V_i,V_j],V_0,...,\hat V_i,...,\hat V_j,...,V_k)}**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [V_i,V_j]}**
denotes Lie bracket and the hat denotes the omission of that element: **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(V_0,...,\hat V_i,...,V_k)=\omega(V_0,..., V_{i-1},V_{i+1}...,V_k).}**

In particular, for 1-forms we have:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\omega(X,Y)=X(\omega(Y))-Y(\omega(X))-\omega([X,Y]).}**

More generally, the Lie derivative is defined via the Lie bracket:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}_XY=[X,Y]}**,

and the Lie derivative of a general differential form is closely related to the exterior derivative. The differences are primarily notational; various identities between the two are provided in the article on Lie derivatives.

## The exterior derivative in 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*: **R**^{n}→**R**, we have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\, dx_i.}**

Therefore, for vector field **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle df(V) = \langle \mbox{grad }f,V\rangle,}**

where *grad f* denotes gradient of *f* and *<•, •> is the scalar product.*

### Curl

For a 1-form **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega=\sum_{i} f_i\,dx_i}**
on **R**^{3},

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \omega=\sum_{i,j}\frac{\partial f_i}{\partial x_j} dx_j\wedge dx_i,}**

which restricted to the three-dimensional case **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega= u\,dx+v\,dy+w\,dz }**
is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \omega = \left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) dx \wedge dy + \left(\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \right) dy \wedge dz + \left(\frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \right) dz \wedge dx.}**

Therefore, for vector fields **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U}**
, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=[u,v,w]}**
and **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W}**
we have
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \omega(U,W)=\langle\mbox{curl}\, V \times U,W\rangle }**
where *curl V* denotes the curl of *V*,
× is the vector product, and *<•, •>* is the scalar product.

### Divergence

For a 2-form **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \sum_{i,j} h_{i,j}\,dx_i\wedge dx_j,}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \omega = \sum_{i,j,k} \frac{\partial h_{i,j}}{\partial x_k} dx_k \wedge dx_i \wedge dx_j.}**

For three dimensions, with **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = p\,dy\wedge dz+q\,dz\wedge dx+r\,dx\wedge dy}**
we get

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \omega\,}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left( \frac{\partial p}{\partial x} + \frac{\partial q}{\partial y} + \frac{\partial r}{\partial z} \right) dx \wedge dy \wedge dz }****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \mbox{div}V\, dx \wedge dy \wedge dz,}**

where *V* is a vector field defined by **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V = [p,q,r].}**

## Examples

For a 1-form **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = u\, dx + v\, dy}**
on **R**^{2} we have

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d \sigma = \left(\frac{\partial{v}}{\partial{x}} - \frac{\partial{u}}{\partial{y}}\right) dx \wedge dy}**

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