Exterior derivative

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In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a form of degree zero, to differential forms of higher degree. Its current form was invented by Élie Cartan.

The exterior derivative d has the property that d2 = 0 and is the differential (coboundary) used to define de Rham (and Alexander-Spanier) cohomology on forms. Integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology of a smooth manifold. The theorem of de Rham shows that this map is actually an isomorphism. In this sense, the exterior derivative is the "dual" of the boundary map on singular simplices.

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[edit] 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\le i_1, i_2, \dots, i_k \le n, the exterior derivative of a k-form

\omega = f_Idx_I=f_{i_1i_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 = ΣI fI dxI (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).

Geometrically, the k + 1 form dω acts on each tangent space of Rn in the following way: a (k + 1)-tuple of vectors (u1,...,uk + 1) in the tangent space defines an oriented (k + 1)-polyhedron p. dω(u1,...,uk + 1) is defined to be the integral of ω over the boundary of p, where the boundary is given the inherited orientation. Assuming the fact that every smooth manifold admits a (smooth) triangulation, this gives immediately Stokes' theorem.

[edit] 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.

[edit] Properties

Exterior differentiation is by definition linear. Direct computation shows that it also has the following properties:

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

It can be shown that the exterior derivative is uniquely determined by these properties and its agreement with the differential on 0-forms (functions).

Differential forms in the kernel of d are said to be closed forms. For instance, a 1-form is closed if on each tangent space, its integral along the boundary of the parallellogram given by any pair of tangent vectors is zero. Thus closedness is a local condition. The the image of d is said to consist of exact forms (cf. exact differentials). It is immediate that exact forms are closed.

The exterior derivative is natural. If f: M → N is a smooth map and Ωk 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*dω, 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 Ωk to Ωk+1.

[edit] Invariant formula

Given a k-form ω and arbitrary smooth vector fields V0,V1, …, Vk 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 [Vi,Vj] 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:

dω(X,Y) = X(ω(Y)) − Y(ω(X)) − ω([X,Y]).

[edit] 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.

[edit] 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.

This is a 1-form, a section of the cotangent bundle, that gives local linear approximation to f on each tangent space.

For a vector field V,

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

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

[edit] Curl

One can associate to a vector field V = (u, v, w) on R3 the 1-form

\omega^1 _V = u dx + v dy + w dz,

and the 2-form

\omega^2 _V = u dy \wedge dz + v dz \wedge dx + w dx \wedge dy.

The integral of ω1V over a path gives work done against -V along the path; locally, it is the dot product with V. The integral of ω2V over a surface gives the flux of V over that surface; locally, it is the scalar triple product with V.

One can check directly that

d \omega^1 _V = \omega^2 _{curl \;V},

where curl V denotes the curl of V. The flux of curl V over a surface is the integral of ω1V over the boundary of the surface.

[edit] Divergence

Similarly,

d \omega^2 _V = \mbox{div}\; V \; dx \wedge dy \wedge dz.

The flux of V over the boundary of a 3-polyhedron p is given by the integral of the divergence of V over p.

[edit] 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.

[edit] See also

[edit] References

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