Exterior derivative

In mathematics, the exterior derivative operator of differential geometry 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 (coboundary) used to define de Rham and Alexander-Spanier cohomology. Its current form was invented by Élie Cartan.

Definition

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

Given a multi-index $I=(i_{1},i_{2},\dots ,i_{k})$ with $1\leq i_{1},i_{2},\dots ,i_{k}\leq n,$ the exterior derivative of a k-form

$\omega =f_{I}dx_{I}=f_{i_{1}i_{2}\cdots i_{k}}dx_{i_{1}}\wedge dx_{i_{2}}\wedge \cdots \wedge dx_{i_{k}}$

over Rⁿ is defined as

$d{\omega }=\sum _{i=1}^{n}{\frac {\partial f_{I}}{\partial x_{i}}}dx_{i}\wedge dx_{I}.$

For general k-forms ω=Σ_I f_I dx_I (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 $dx_{i}\wedge dx_{I}=0$ (see wedge product).

Examples

For a 1-form $\sigma =u\,dx+v\,dy$ on R² we have, by applying the above formula to each term,

$d\sigma =\left({\frac {\partial {u}}{\partial {x}}}dx\wedge dx+{\frac {\partial {u}}{\partial {y}}}dy\wedge dx\right)+\left({\frac {\partial {v}}{\partial {x}}}dx\wedge dy+{\frac {\partial {v}}{\partial {y}}}dy\wedge dy\right)$

$=0-{\frac {\partial {u}}{\partial {y}}}dx\wedge dy+{\frac {\partial {v}}{\partial {x}}}dx\wedge dy+0$

$=\left({\frac {\partial {v}}{\partial {x}}}-{\frac {\partial {u}}{\partial {y}}}\right)dx\wedge dy.$

Properties

Exterior differentiation satisfies three important properties:

• linearity

• the wedge product rule (see antiderivation)

$d(\omega \wedge \eta )=d\omega \wedge \eta +(-1)^{{\rm {deg\,}}\omega }(\omega \wedge d\eta )$

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

$d^{2}(\omega )=d(d\omega )=0\,\!$

It can be shown that the 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).

The exterior derivative is natural. If f: M → N is a smooth map and Ω^k and Ω^k+1 are the contravariant smooth functors that assign correspondingly to each manifold the space of k- and k+1-forms on the manifold, then the following diagram commutes

so d(f*ω) = f*dω, where f* denotes the pullback of f. Thus d is a natural transformation from Ω^k to Ω^k+1.

Invariant formula

Given a k-form ω and arbitrary smooth vector fields V₀,V₁, …, V_k we have

$d\omega (V_{0},V_{1},...V_{k})=\sum _{i}(-1)^{i}V_{i}\left(\omega (V_{0},\ldots ,{\hat {V}}_{i},\ldots ,V_{k})\right)$

$+\sum _{i<j}(-1)^{i+j}\omega ([V_{i},V_{j}],V_{0},\ldots ,{\hat {V}}_{i},\ldots ,{\hat {V}}_{j},\ldots ,V_{k})$

where $[V_{i},V_{j}]$ denotes Lie bracket and the hat denotes the omission of that element: $\omega (V_{0},\ldots ,{\hat {V}}_{i},\ldots ,V_{k})=\omega (V_{0},\ldots ,V_{i-1},V_{i+1},\ldots ,V_{k}).$

In particular, for 1-forms we have:

$d\omega (X,Y)=X(\omega (Y))-Y(\omega (X))-\omega ([X,Y]).$

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: Rⁿ→R, we have

$df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}\,dx_{i}.$

Therefore, for vector field $V$

$df(V)=\langle {\mbox{grad }}f,V\rangle ,$

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

Curl

For a 1-form $\omega =\sum _{i}f_{i}\,dx_{i}$,

$d\omega =\sum _{i,j}{\frac {\partial f_{i}}{\partial x_{j}}}dx_{j}\wedge dx_{i},$

or

$d\omega =\sum _{i<j}\left({\frac {\partial f_{j}}{\partial x_{i}}}-{\frac {\partial f_{i}}{\partial x_{j}}}\right)dx_{i}\wedge dx_{j}$

which restricted to the three-dimensional case $\omega =u\,dx+v\,dy+w\,dz$ is

$d\omega =\left({\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}\right)dx\wedge dy+\left({\frac {\partial w}{\partial y}}-{\frac {\partial v}{\partial z}}\right)dy\wedge dz+\left({\frac {\partial u}{\partial z}}-{\frac {\partial w}{\partial x}}\right)dz\wedge dx.$

Therefore, for vector fields $U$, $V=[u,v,w]$ and $W$ we have $d\omega (U,W)=\langle {\mbox{curl}}\,V\times U,W\rangle$ where curl V denotes the curl of V, × is the vector product, and < , > is the scalar product.

Divergence

For a 2-form $\omega =\sum _{i,j}h_{i,j}\,dx_{i}\wedge dx_{j},$

$d\omega =\sum _{i,j,k}{\frac {\partial h_{i,j}}{\partial x_{k}}}dx_{k}\wedge dx_{i}\wedge dx_{j}.$

For three dimensions, with $\omega =p\,dy\wedge dz+q\,dz\wedge dx+r\,dx\wedge dy$ we get

$d\omega \,$	$=\left({\frac {\partial p}{\partial x}}+{\frac {\partial q}{\partial y}}+{\frac {\partial r}{\partial z}}\right)dx\wedge dy\wedge dz$
	$={\mbox{div}}V\,dx\wedge dy\wedge dz,$

where V is a vector field defined by $V=[p,q,r].$

Invariant formulations of div, grad, and curl

The three operators above can be written in coordinate-free notation as follows:

${\begin{array}{rcl}\nabla f&=&\left({\mathbf {d} }f\right)^{\sharp }\\\nabla \times F&=&\left[\star \left({\mathbf {d} }F^{\flat }\right)\right]^{\sharp }\\\nabla \cdot F&=&\star {\mathbf {d} }\left(\star F^{\flat }\right)\\\end{array}}$

where $\star$ is the Hodge star operator and $\flat$ and $\sharp$ are the musical isomorphisms.

See also

• Exterior covariant derivative
• Green's theorem
• Lie derivative
• Stokes' theorem

References

• Flanders, Harley (1989). Differential forms with applications to the physical sciences. New York: Dover Publications. pp. page 20. ISBN 0-486-66169-5. {{cite book}}: |pages= has extra text (help)
• Ramanan, S. (2005). Global calculus. Providence, Rhode Island: American Mathematical Society. pp. page 54. ISBN 0-8218-3702-8. {{cite book}}: |pages= has extra text (help)
• Conlon, Lawrence (2001). Differentiable manifolds. Basel, Switzerland: Birkhäuser. pp. page 239. ISBN 0-8176-4134-3. {{cite book}}: |pages= has extra text (help)
• Darling, R. W. R. (1994). Differential forms and connections. Cambridge, UK: Cambridge University Press. pp. page 35. ISBN 0-521-46800-0. {{cite book}}: |pages= has extra text (help)