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[edit]

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, df is the unique 1-form such that for every smooth vector field X, df (X) = dXf, where dXf 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[edit]

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

  1. df is the differential of f for smooth functions f.
  2. d(df ) = 0 for any smooth function f.
  3. d(αβ) = dαβ + (−1)p (α ∧ dβ) where α 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(dα) = 0 for any k-form α; more succinctly, d2 = 0. The third defining property implies as a special case that if f is a function and α a k-form, then d() = d(f  ∧ α) = df  ∧ α +  f  ∧ dα 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[edit]

Alternatively, one can work entirely in a local coordinate system (x1, ..., xn). First, the coordinate differentials dx1, ..., dxn 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 I = (i1, ..., ik) with 1 ≤ ipn for 1 ≤ pk, the exterior derivative of a k-form

\omega = f_I \mathrm{d} x^I = f_{i_1,i_2\cdots i_k}\mathrm{d}x^{i_1}\wedge \mathrm{d}x^{i_2}\wedge\cdots\wedge \mathrm{d}x^{i_k}

over Rn is defined as

\mathrm{d}{\omega} = \sum_{i=1}^n \frac{\partial f_I}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d} x^I.

For a general k-form

\omega = \sum_I f_I \mathrm{d}x^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 dxi ∧ dxI = 0 (see wedge product).

The definition of the exterior derivative in local coordinates follows from the preceding definition. Indeed, if ω =  fI dxi1 ∧ ... ∧ dxik, then

\begin{align}
\mathrm{d}{\omega} &= \mathrm{d} \left (f_I \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \right ) \\
                   &= \mathrm{d}f_I  \wedge \left (\mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \right ) + f_I \mathrm{d} \left (
                      \mathrm{d} x^{i_1}\wedge \cdots \wedge \mathrm{d}x^{i_k} \right ) \\
                   &= \mathrm{d}f_I \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} + \sum_{p=1}^k (-1)^{(p-1)} f_I \mathrm{d} x^{i_1}
                      \wedge \cdots \wedge \mathrm{d}x^{i_{p-1}} \wedge \mathrm{d}^2x^{i_p} \wedge \mathrm{d}x^{i_{p+1}} \wedge \cdots \wedge\mathrm{d}
                      x^{i_k} \\
                     &= \mathrm{d}f_I \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \\
                     &= \sum_{i=1}^n \frac{\partial f_I}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_k} \\
\end{align}

Here, we have interpreted fI as a 0-form, and then applied the properties of the exterior derivative.

Invariant formula[edit]

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 V0,V1, ..., Vk:

\mathrm{d}\omega(V_0,...,V_k) = \sum_i(-1)^{i} V_i \left( \omega \left (V_0, \ldots, \hat V_i, \ldots,V_k \right )\right) +\sum_{i<j}(-1)^{i+j}\omega \left (\left [V_i, V_j \right ], V_0, \ldots, \hat V_i, \ldots, \hat V_j, \ldots, V_k \right )

where [Vi, Vj] denotes the Lie bracket and a hat denotes the omission of that element:

\omega \left (V_0, \ldots, \hat V_i, \ldots,V_k \right ) = \omega \left (V_0, \ldots, V_{i-1}, V_{i+1}, \ldots, V_k \right ).

In particular, for 1-forms we have: dω(X, Y) = (Y) − (X) − ω([X, Y]), where X and Y are vector fields.

Stokes' Theorem on manifolds[edit]

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:

\int_M \mathrm{d}\omega = \int_{\partial{M}} \omega

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[edit]

Example 1. Consider σ = u dx1 ∧ dx2 over a 1-form basis dx1, ..., dxn. The exterior derivative is:

\begin{align}
  \mathrm{d} \sigma &= \mathrm{d}(u) \wedge \mathrm{d}x^1 \wedge \mathrm{d}x^2 \\
                    &= \left(\sum_{i=1}^n \frac{\partial u}{\partial x^i} \mathrm{d}x^i\right) \wedge \mathrm{d}x^1 \wedge \mathrm{d}x^2 \\
                    &= \sum_{i=3}^n \left( \frac{\partial u}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}x^1 \wedge \mathrm{d}x^2 \right )
\end{align}

The last formula follows easily from the properties of the wedge product. Namely, dxi ∧ dxi = 0.

Example 2. Let σ = u dx + v dy be a 1-form defined over R2. By applying the above formula to each term (consider x1 = x and x2 = y) we have the following sum,

\begin{align}
\mathrm{d} \sigma 
    &= \left( \sum_{i=1}^2 \frac{\partial u}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}x \right) + \left( \sum_{i=1}^2 \frac{\partial v}{\partial x^i} \mathrm{d}x^i \wedge \mathrm{d}y \right) \\
    &= \left(\frac{\partial{u}}{\partial{x}} \mathrm{d}x \wedge \mathrm{d}x + \frac{\partial{u}}{\partial{y}} \mathrm{d}y \wedge \mathrm{d}x\right) + \left(\frac{\partial{v}}{\partial{x}} \mathrm{d}x \wedge \mathrm{d}y + \frac{\partial{v}}{\partial{y}} \mathrm{d}y \wedge \mathrm{d}y\right) \\
    &= 0 - \frac{\partial{u}}{\partial{y}} \mathrm{d}x \wedge \mathrm{d}y + \frac{\partial{v}}{\partial{x}} \mathrm{d}x \wedge \mathrm{d}y + 0 \\
    &= \left(\frac{\partial{v}}{\partial{x}} - \frac{\partial{u}}{\partial{y}}\right) \mathrm{d}x \wedge \mathrm{d}y
\end{align}

Further properties[edit]

Closed and exact forms[edit]

A k-form ω is called closed if dω = 0; closed forms are the kernel of d. ω is called exact if ω = dα for some (k − 1)-form α; exact forms are the image of d. Because d2 = 0, every exact form is closed. The Poincaré lemma states that in a contractible region, the converse is true.

de Rham cohomology[edit]

Because the exterior derivative d has the property that d2 = 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[edit]

The exterior derivative is natural in the technical sense: if f : MN 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

Exteriorderivnatural.png

so d(fω) =  fdω, where f denotes the pullback of f. This follows from that fω(·), by definition, is ω(f(·)), f being the pushforward of f. Thus d is a natural transformation from Ωk to Ωk+1.

Exterior derivative in vector calculus[edit]

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

Gradient[edit]

A smooth function f : RnR is a 0-form. The exterior derivative of this 0-form is the 1-form

\mathrm{d}f = \sum_{i=1}^n \frac{\partial f}{\partial x^i}\, \mathrm{d}x^i = \langle \nabla f,\cdot \rangle.

That is, the form df 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 df is a section of the cotangent bundle, that gives a local linear approximation to f in the cotangent space at each point.

Divergence[edit]

A vector field V = (v1, v2, ... vn) on Rn has a corresponding (n − 1)-form

\begin{align}
\omega_V &= v_1 \left (\mathrm{d}x^2 \wedge \mathrm{d}x^3 \wedge \cdots \wedge \mathrm{d}x^n \right) - v_2 \left (\mathrm{d}x^1 \wedge \mathrm{d}x^3  \cdots \wedge \mathrm{d}x^n \right ) + \cdots + (-1)^{n-1}v_n \left (\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^{n-1} \right) \\
&=\sum_{p=1}^n (-1)^{(p-1)}v_p \left (\mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^{p-1} \wedge \widehat{\mathrm{d}x^{p}} \wedge \mathrm{d}x^{p+1} \wedge \cdots \wedge \mathrm{d}x^n \right )
\end{align}

where \widehat{\mathrm{d}x^{p}} denotes the omission of that element.

(For instance, when n = 3, in three-dimensional space, the 2-form ωV is locally the scalar triple product with V.) The integral of ωV over a hypersurface is the flux of V over that hypersurface.

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

\mathrm{d} \omega _V = \operatorname{div}(V) \left (\mathrm{d}x^1 \wedge \mathrm{d}x^2 \wedge \cdots \wedge \mathrm{d}x^n \right ).

Curl[edit]

A vector field V on Rn also has a corresponding 1-form

\eta_V = v_1  \mathrm{d}x^1 + v_2 \mathrm{d}x^2 + \cdots + v_n \mathrm{d}x^n.,

Locally, ηV is the dot product with V. The integral of ηV along a path is the work done against V along that path.

When n = 3, in three-dimensional space, the exterior derivative of the 1-form ηV is the 2-form

\mathrm{d} \eta_V = \omega_{\operatorname{curl}(V)}.

Invariant formulations of grad, curl, div, and Laplacian[edit]

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


\begin{array}{rcccl}
      \operatorname{grad}(f) &=& \nabla f        &=& \left( \mathrm{d} f \right)^\sharp \\
      \operatorname{div}(F)  &=& \nabla \cdot F  &=& \star \mathrm{d} \left( \star F^\flat \right) \\
      \operatorname{curl}(F) &=& \nabla \times F &=& \left[ \star \left( \mathrm{d} F^\flat \right) \right]^\sharp, \\
      \Delta f               &=& \nabla^2 f      &=& \star \mathrm{d} \left( \star \mathrm{d} f \right) \\
\end{array}

where \star is the Hodge star operator and \flat and \sharp are the musical isomorphisms. Since these isomorphisms incorporate the Riemannian metric, these operators take into account the curved geometry of the manifold, i.e., they are not (in general) the same as div, grad, and curl on Euclidean Rn.

See also[edit]

References[edit]

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