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 in 1899. The resulting calculus, known as exterior calculus, allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.

If a differential Template:Math-form is thought of as measuring the flux through an infinitesimal Template:Math-parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a Template:Math-parallelotope at each point.

Definition

The exterior derivative of a differential form of degree Template:Math (also differential Template:Math-form, or just Template:Math-form for brevity here) is a differential form of degree Template:Math.

If Template:Math is a smooth function (a Template:Math-form), then the exterior derivative of Template:Math is the differential of Template:Math. That is, Template:Math is the unique [[1-form|Template:Math-form]] such that for every smooth vector field Template:Math, Template:Math, where Template:Math is the directional derivative of Template:Math in the direction of Template:Math.

The exterior product of differential forms (denoted with the same symbol Template:Math) is defined as their pointwise exterior product.

There are a variety of equivalent definitions of the exterior derivative of a general Template:Math-form.

In terms of axioms

The exterior derivative is defined to be the unique Template:Math-linear mapping from Template:Math-forms to Template:Math-forms that has the following properties:

• The operator <math>d</math> applied to the <math>0</math>-form <math>f</math> is the differential <math>df</math> of <math>f</math>
• If <math>\alpha</math> and <math>\beta</math> are two <math>k</math>-forms, then <math>d(a\alpha+b\beta)=ad\alpha+bd\beta</math> for any field elements <math>a,b</math>
• If <math>\alpha</math> is a <math>k</math>-form and <math>\beta</math> is an <math>l</math>-form, then <math>d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^k\alpha\wedge d\beta</math> (graded product rule)
• If <math>\alpha</math> is a <math>k</math>-form, then <math>d(d\alpha)=0</math> (Poincaré's lemma)

If <math>f</math> and <math>g</math> are two <math>0</math>-forms (functions), then from the third property for the quantity <math>d(f\wedge g)</math>, which is simply <math>d(fg)</math>, the familiar product rule <math>d(fg)=g\,df+f\,dg</math> is recovered. The third property can be generalised, for instance, if <math>\alpha</math> is a <math>k</math>-form, <math>\beta</math> is an <math>l</math>-form and <math>\gamma</math> is an <math>m</math>-form, then

<math>d(\alpha\wedge\beta\wedge\gamma)=d\alpha\wedge\beta\wedge\gamma+(-1)^k\alpha\wedge d\beta\wedge\gamma+(-1)^{k+l}\alpha\wedge\beta\wedge d\gamma.</math>

In terms of local coordinates

Alternatively, one can work entirely in a local coordinate system Template:Math. The coordinate differentials Template:Math form a basis of the space of one-forms, each associated with a coordinate. Given a multi-index Template:Math with Template:Math for Template:Math (and denoting Template:Math with Template:Math), the exterior derivative of a (simple) Template:Math-form

<math>\varphi = g\,dx^I = g\,dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k}</math>

over Template:Math is defined as

<math>d{\varphi} = dg\wedge dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k} = \frac{\partial g}{\partial x^j} \, dx^j \wedge \,dx^{i_1}\wedge dx^{i_2}\wedge\cdots\wedge dx^{i_k}</math>

(using the Einstein summation convention). The definition of the exterior derivative is extended linearly to a general Template:Math-form (which is expressible as a linear combination of basic simple <math>k</math>-forms)

<math>\omega = f_I \, dx^I,</math>

where each of the components of the multi-index Template:Math run over all the values in Template:Math. Note that whenever Template:Math equals one of the components of the multi-index Template:Math then Template:Math (see Exterior product).

The definition of the exterior derivative in local coordinates follows from the preceding definition in terms of axioms. Indeed, with the Template:Math-form Template:Math as defined above,

<math>\begin{align}

 d{\varphi} &= d\left (g\,dx^{i_1} \wedge \cdots \wedge dx^{i_k} \right ) \\
            &= dg \wedge \left (dx^{i_1} \wedge \cdots \wedge dx^{i_k} \right ) +
                 g\,d\left (dx^{i_1}\wedge \cdots \wedge dx^{i_k} \right ) \\
            &= dg \wedge dx^{i_1} \wedge \cdots \wedge dx^{i_k} + g \sum_{p=1}^k (-1)^{p-1} \, dx^{i_1}
                  \wedge \cdots \wedge dx^{i_{p-1}} \wedge d^2x^{i_p} \wedge dx^{i_{p+1}} \wedge \cdots \wedge dx^{i_k} \\
            &= dg \wedge dx^{i_1} \wedge \cdots \wedge dx^{i_k} \\
            &= \frac{\partial g}{\partial x^i} \, dx^i \wedge dx^{i_1} \wedge \cdots \wedge dx^{i_k} \\

\end{align}</math>

Here, we have interpreted Template:Math as a Template:Math-form, and then applied the properties of the exterior derivative.

This result extends directly to the general Template:Math-form Template:Math as

<math>d\omega = \frac{\partial f_I}{\partial x^i} \, dx^i \wedge dx^I .</math>

In particular, for a Template:Math-form Template:Math, the components of Template:Math in local coordinates are

<math>(d\omega)_{ij} = \partial_i \omega_j - \partial_j \omega_i. </math>

Caution: There are two conventions regarding the meaning of <math>dx^{i_1} \wedge \cdots \wedge dx^{i_k}</math>. Most current authorsTemplate:Fact have the convention that

<math>\left(dx^{i_1} \wedge \cdots \wedge dx^{i_k}\right) \left( \frac{\partial}{\partial x^{i_1}}, \ldots, \frac{\partial}{\partial x^{i_k}} \right) = 1 .</math>

while in older texts like Kobayashi and Nomizu or Helgason

<math>\left(dx^{i_1} \wedge \cdots \wedge dx^{i_k}\right) \left( \frac{\partial}{\partial x^{i_1}}, \ldots, \frac{\partial}{\partial x^{i_k}} \right) = \frac{1}{k!} .</math>

In terms of invariant formula

Alternatively, an explicit formula can be given <ref> Spivak(1970), p 7-18, Th. 13 </ref> for the exterior derivative of a Template:Math-form Template:Math, when paired with Template:Math arbitrary smooth vector fields Template:Math:

<math>d\omega(V_0, \ldots, V_k) = \sum_i(-1)^{i} V_i ( \omega (V_0, \ldots, \widehat V_i, \ldots,V_k )) + \sum_{i<j}(-1)^{i+j}\omega ([V_i, V_j], V_0, \ldots, \widehat V_i, \ldots, \widehat V_j, \ldots, V_k )</math>

where Template:Math denotes the Lie bracket and a hat denotes the omission of that element:

<math>\omega (V_0, \ldots, \widehat V_i, \ldots, V_k ) = \omega(V_0, \ldots, V_{i-1}, V_{i+1}, \ldots, V_k ).</math>

In particular, when Template:Math is a Template:Math-form we have that Template:Math.

Note: With the conventions of e.g., Kobayashi–Nomizu and Helgason the formula differs by a factor of Template:Math:

<math>\begin{align}

 d\omega(V_0, \ldots, V_k) ={}
   & {1 \over k+1} \sum_i(-1)^i \, V_i ( \omega (V_0, \ldots, \widehat V_i, \ldots,V_k  )) \\
   & {}+ {1 \over k+1}  \sum_{i<j}(-1)^{i+j}\omega([V_i, V_j], V_0, \ldots, \widehat V_i, \ldots, \widehat V_j, \ldots, V_k ).

\end{align}</math>

Examples

Example 1. Consider Template:Math over a Template:Math-form basis Template:Math for a scalar field Template:Math. The exterior derivative is:

<math>\begin{align}

 d\sigma &= du \wedge dx^1 \wedge dx^2 \\
                   &= \left(\sum_{i=1}^n \frac{\partial u}{\partial x^i} \, dx^i\right) \wedge dx^1 \wedge dx^2 \\
                   &= \sum_{i=3}^n \left( \frac{\partial u}{\partial x^i} \, dx^i \wedge dx^1 \wedge dx^2 \right )

\end{align}</math>

The last formula, where summation starts at Template:Math, follows easily from the properties of the exterior product. Namely, Template:Math.

Example 2. Let Template:Math be a Template:Math-form defined over Template:Math. By applying the above formula to each term (consider Template:Math and Template:Math) we have the sum

<math>\begin{align}

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

\end{align}</math>

Stokes' theorem on manifolds

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If Template:Math is a compact smooth orientable Template:Math-dimensional manifold with boundary, and Template:Math is an Template:Math-form on Template:Math, then the generalized form of Stokes' theorem states that

<math>\int_M d\omega = \int_{\partial{M}} \omega</math>

Intuitively, if one thinks of Template:Math 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 Template:Math.

Further properties

Closed and exact forms

Template:Main article A Template:Math-form Template:Math is called closed if Template:Math; closed forms are the kernel of Template:Math. Template:Math is called exact if Template:Math for some Template:Math-form Template:Math; exact forms are the image of Template:Math. Because Template:Math, every exact form is closed. The Poincaré lemma states that in a contractible region, the converse is true.

de Rham cohomology

Because the exterior derivative Template:Math has the property that Template:Math, it can be used as the differential (coboundary) to define de Rham cohomology on a manifold. The Template:Math-th de Rham cohomology (group) is the vector space of closed Template:Math-forms modulo the exact Template:Math-forms; as noted in the previous section, the Poincaré lemma states that these vector spaces are trivial for a contractible region, for Template:Math. For smooth manifolds, integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology over Template:Math. 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

The exterior derivative is natural in the technical sense: if Template:Math is a smooth map and Template:Math is the contravariant smooth functor that assigns to each manifold the space of Template:Math-forms on the manifold, then the following diagram commutes

File:Exteriorderivnatural.png

so Template:Math, where Template:Math denotes the pullback of Template:Math. This follows from that Template:Math, by definition, is Template:Math, Template:Math being the pushforward of Template:Math. Thus Template:Math is a natural transformation from Template:Math to Template:Math.

Exterior derivative in vector calculus

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

Gradient

A smooth function Template:Math on a real differentiable manifold Template:Math is a Template:Math-form. The exterior derivative of this Template:Math-form is the Template:Math-form Template:Math.

When an inner product Template:Math is defined, the gradient Template:Math of a function Template:Math is defined as the unique vector in Template:Math such that its inner product with any element of Template:Math is the directional derivative of Template:Math along the vector, that is such that

<math>\langle \nabla f, \cdot \rangle = df = \sum_{i=1}^n \frac{\partial f}{\partial x^i}\, dx^i .</math>

That is,

<math>\nabla f = (df)^\sharp = \sum_{i=1}^n \frac{\partial f}{\partial x^i}\, \left(dx^i\right)^\sharp ,</math>

where Template:Math denotes the musical isomorphism Template:Math mentioned earlier that is induced by the inner product.

The Template:Math-form Template:Math is a section of the cotangent bundle, that gives a local linear approximation to Template:Math in the cotangent space at each point.

Divergence

A vector field Template:Math on Template:Math has a corresponding Template:Math-form

<math>\begin{align}

 \omega_V &= v_1 \left (dx^2 \wedge \cdots \wedge dx^n \right) - v_2 \left (dx^1 \wedge dx^3 \wedge \cdots \wedge dx^n \right ) + \cdots + (-1)^{n-1}v_n \left (dx^1 \wedge \cdots \wedge dx^{n-1} \right) \\
          &= \sum_{i=1}^n (-1)^{(i-1)}v_i \left (dx^1 \wedge \cdots \wedge dx^{i-1} \wedge \widehat{dx^{i}} \wedge dx^{i+1} \wedge \cdots \wedge dx^n \right )

\end{align}</math> where <math>\widehat{dx^{i}}</math> denotes the omission of that element.

(For instance, when Template:Math, i.e. in three-dimensional space, the Template:Math-form Template:Math is locally the scalar triple product with Template:Math.) The integral of Template:Math over a hypersurface is the flux of Template:Math over that hypersurface.

The exterior derivative of this Template:Math-form is the Template:Math-form

<math>d\omega _V = \operatorname{div} V \left (dx^1 \wedge dx^2 \wedge \cdots \wedge dx^n \right ).</math>

Curl

A vector field Template:Math on Template:Math also has a corresponding Template:Math-form

<math>\eta_V = v_1 \, dx^1 + v_2 \, dx^2 + \cdots + v_n \, dx^n.</math>

Locally, Template:Math is the dot product with Template:Math. The integral of Template:Math along a path is the work done against Template:Math along that path.

When Template:Math, in three-dimensional space, the exterior derivative of the Template:Math-form Template:Math is the Template:Math-form

<math>d\eta_V = \omega_{\operatorname{curl} V}.</math>

Invariant formulations of operators in vector calculus

The standard vector calculus operators can be generalized for any pseudo-Riemannian manifold, and written in coordinate-free notation as follows:

<math>\begin{array}{rcccl}

 \operatorname{grad} f &\equiv& \nabla f        &=& \left( d f \right)^\sharp \\
 \operatorname{div} F  &\equiv& \nabla \cdot F  &=& {\star d {\star} \mathord{\left( F^\flat \right)}} \\
 \operatorname{curl} F &\equiv& \nabla \times F &=& \left( {\star} d \mathord{\left( F^\flat \right)} \right)^\sharp \\
 \Delta f              &\equiv& \nabla^2 f      &=& {\star} d {\star} d f \\
                       &      & \nabla^2 F      &=& \left(d{\star}d{\star}\mathord{\left(F^{\flat}\right)} - {\star}d{\star}d\mathord{\left(F^{\flat}\right)}\right)^{\sharp} , \\

\end{array}</math> where Template:Math is the Hodge star operator, Template:Math and Template:Math are the musical isomorphisms, Template:Math is a scalar field and Template:Math is a vector field.

Note that the expression for Template:Math requires Template:Math to act on Template:Math, which is a form of degree Template:Math. A natural generalization of Template:Math to Template:Math-forms of arbitrary degree allows this expression to make sense for any Template:Math.

See also

• Exterior covariant derivative
• de Rham complex
• Finite element exterior calculus
• Discrete exterior calculus
• Green's theorem
• Lie derivative
• Stokes' theorem
• Fractal derivative

References

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