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.

[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

$\omega = f_Idx_I=f_{i_1i_2\cdots i_k}dx_{i_1}\wedge dx_{i_2}\wedge\cdots\wedge dx_{i_k}$

where

$dx_I = dx_{i_1}\wedge dx_{i_2}\wedge\cdots\wedge dx_{i_k}$

$f_I = f_{i_1i_2\cdots i_k}$

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).

Geometrically, the k + 1 form dω acts on each tangent space of Rⁿ in the following way: a (k + 1)-tuple of vectors (u₁,...,u_{k + 1}) in the tangent space defines an oriented (k + 1)-polyhedron p. Then dω(u₁,...,u_{k + 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] Alternative definition

The exterior derivative of a differential form of degree k is again a differential form of degree k + 1, with the following properties:

$df = \nabla f$ for smooth functions (zero-forms)
$d (x + y) = d x + d y$
$d\,dx = 0$
$d(x \wedge y) = dx \wedge y + (-1)^{k} x \wedge dy$

we can easily obtain the previous definition:

$d{\omega} = d ( f_I dx_{i_1} \wedge \cdots \wedge dx_{i_k} ) = d \left ( \frac{1}{k!} f_I \wedge dx_{i_1} \wedge \cdots \wedge dx_{i_k} \right)$

$= \frac{1}{k!}d(f_I) \wedge (dx_{i_1} \wedge \cdots \wedge dx_{i_k}) + (-1)^{p} d(dx_{i_1} \wedge \cdots \wedge dx_{i_k})$

$= \frac{1}{k!}d(f_I) \wedge (dx_{i_1} \wedge \cdots \wedge dx_{i_k})$

$= \frac{1}{k!}\frac{\partial(f_I)}{\partial x^n} dx_n \wedge (dx_{i_1} \wedge \cdots \wedge dx_{i_k})$

we have here interpreted f_I as a zero-form, and then applied the properties of the exterior derivative.

[edit] Examples

[edit] 1

Consider $\sigma = u\, dx_1 \wedge dx_2$ over a 1-form basis $dx_1, dx_2, \ldots, dx_n$ . The exterior derivative is:

$d \sigma = \sum_{i=1}^n \frac{\partial u}{\partial x_i} dx_i \wedge dx_1 \wedge dx_2$

$= \sum_{i=3}^n \frac{\partial u}{\partial x_i} dx_i \wedge dx_1 \wedge dx_2$

The last formula follows easily from the properties of the wedge product.

[edit] 2

For a 1-form $\sigma = u\, dx + v\, dy$ on R² we have, by applying the above formula to each term (consider $x 1 = x$ and $x 2 = y$ in the following sum),

$d \sigma = \left( \sum_{i=1}^2 \frac{\partial u}{\partial x_i} dx_i \wedge dx \right) + \left( \sum_{i=1}^2 \frac{\partial v}{\partial x_i} dx_i \wedge dy \right)$

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

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

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

[edit] Properties

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

the wedge product rule holds (see antiderivation)

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

and d² = 0, which follows from the equality of mixed partial derivatives.

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 parallelogram given by any pair of tangent vectors is zero. Thus closedness is a local condition. 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 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*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 Ω^k to Ω^k+1.

[edit] 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 ω(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 ω¹_V over a path gives work done against -V along the path; locally, it is the dot product with V. The integral of ω²_V 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 ω¹_V 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 grad, curl, and div

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. pp. 20. ISBN 0-486-66169-5.
Ramanan, S. (2005). Global calculus. Providence, Rhode Island: American Mathematical Society. pp. 54. ISBN 0-8218-3702-8.
Conlon, Lawrence (2001). Differentiable manifolds. Basel, Switzerland: Birkhäuser. pp. 239. ISBN 0-8176-4134-3.
Darling, R. W. R. (1994). Differential forms and connections. Cambridge, UK: Cambridge University Press. pp. 35. ISBN 0-521-46800-0.

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