HOME
TheInfoList



On a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an at ...
, the exterior derivative extends the concept of the differential of a function to
differential form In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curve ...
s of higher degree. The exterior derivative was first described in its current form by
Élie Cartan Élie Joseph Cartan, ForMemRS (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential g ...
in 1899. It allows for a natural, metric-independent generalization of
Stokes' theorem An illustration of Stokes' theorem, with surface , its boundary and the normal vector . Stokes' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fuji ...
,
Gauss's theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclos ...
, and
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively oriented, ...
from vector calculus. If a differential -form is thought of as measuring the
flux of \mathbf(\mathbf) with the unit normal vector \mathbf(\mathbf) ''(blue arrows)'' at the point \mathbf multiplied by the area dS. The sum of \mathbf\cdot\mathbf dS for each patch on the surface is the flux through the surface Flux describes ...
through an infinitesimal - 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 -parallelotope at each point.


Definition

The exterior derivative of a
differential form In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curve ...
of degree (also differential -form, or just -form for brevity here) is a differential form of degree . If is a
smooth function is a smooth function with compact support. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered "smoot ...
(a -form), then the exterior derivative of is the differential of . That is, is the unique -form such that for every smooth
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attach ...
, , where is the
directional derivative In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v ...
of in the direction of . The exterior product of differential forms (denoted with the same symbol ) is defined as their
pointwiseIn mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined o ...
exterior product In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which are disjoint from S. It is itself an open set and is disjoint from S. The exterior of S in X is often denoted by \operatorname_X S or, if X is ...
. There are a variety of equivalent definitions of the exterior derivative of a general -form.


In terms of axioms

The exterior derivative is defined to be the unique -linear mapping from -forms to -forms that has the following properties: # is the differential of for a -form . # for a -form . # where is a -form. That is to say, is an antiderivation of degree on the
exterior algebra In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, ...
of differential forms. The second defining property holds in more generality: for any -form ; more succinctly, . The third defining property implies as a special case that if is a function and a is -form, then because a function is a -form, and scalar multiplication and the exterior product are equivalent when one of the arguments is a scalar.


In terms of local coordinates

Alternatively, one can work entirely in a
local coordinate system In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an at ...
. The coordinate differentials form a basis of the space of one-forms, each associated with a coordinate. Given a
multi-index Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices. ...
with for (and denoting with an
abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors and ...
), the exterior derivative of a (simple) -form :\varphi = g\,dx^I = g\,dx^\wedge dx^\wedge\cdots\wedge dx^ over is defined as :d = \frac \, dx^i \wedge dx^I (using the
Einstein summation convention In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational br ...
). The definition of the exterior derivative is extended
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics include the linear relationship of voltage and cu ...
ly to a general -form :\omega = f_I \, dx^I, where each of the components of the multi-index run over all the values in . Note that whenever equals one of the components of the multi-index then (see ''
Exterior product In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which are disjoint from S. It is itself an open set and is disjoint from S. The exterior of S in X is often denoted by \operatorname_X S or, if X is ...
''). The definition of the exterior derivative in local coordinates follows from the preceding definition in terms of axioms. Indeed, with the -form as defined above, :\begin d &= d\left (g\,dx^ \wedge \cdots \wedge dx^ \right ) \\ &= dg \wedge \left (dx^ \wedge \cdots \wedge dx^ \right ) + g\,d\left ( dx^\wedge \cdots \wedge dx^ \right ) \\ &= dg \wedge dx^ \wedge \cdots \wedge dx^ + g \sum_^k (-1)^ \, dx^ \wedge \cdots \wedge dx^ \wedge d^2x^ \wedge dx^ \wedge \cdots \wedge dx^ \\ &= dg \wedge dx^ \wedge \cdots \wedge dx^ \\ &= \frac \, dx^i \wedge dx^ \wedge \cdots \wedge dx^ \\ \end Here, we have interpreted as a -form, and then applied the properties of the exterior derivative. This result extends directly to the general -form as :d\omega = \frac \, dx^i \wedge dx^I . In particular, for a -form , the components of in
local coordinates Local coordinates are the ones used in a ''local coordinate system'' or a ''local coordinate space''. Simple examples: * Houses. In order to work in a house construction, the measurements are referred to a control arbitrary point that will allow t ...
are :(d\omega)_ = \partial_i \omega_j - \partial_j \omega_i. ''Caution'': There are two conventions regarding the meaning of dx^ \wedge \cdots \wedge dx^. Most current authors have the convention that :(dx^ \wedge \cdots \wedge dx^) \left( \frac,\ldots, \frac \right) = 1 . while in older text like Kobayashi and Nomizu or Helgason :(dx^ \wedge \cdots \wedge dx^) \left( \frac,\ldots, \frac \right) = \frac 1 .


In terms of invariant formula

Alternatively, an explicit formula can be given for the exterior derivative of a -form , when paired with arbitrary smooth
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attach ...

vector field
s : :d\omega(V_0,...,V_k) = \sum_i(-1)^ d_ \left( \omega \left (V_0, \ldots, \hat V_i, \ldots,V_k \right )\right) +\sum_(-1)^\omega \left (\left _i, V_j \right V_0, \ldots, \hat V_i, \ldots, \hat V_j, \ldots, V_k \right ) where denotes the
Lie bracket In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, \ (x, y) \mapsto , y/math>, t ...
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_, V_, \ldots, V_k \right ). In particular, when is a -form we have that . Note: With the conventions of e.g., Kobayashi–Nomizu and Helgason the formula differs by a factor of : :\begin d\omega(V_0,\ldots,V_k) = & \sum_i(-1)^i \, d_ \left( \omega \left (V_0, \ldots, \hat V_i, \ldots,V_k \right ) \right) \\ & + \sum_(-1)^\omega \left( _i, V_j V_0, \ldots, \hat V_i, \ldots, \hat V_j, \ldots, V_k \right). \end


Examples

Example 1. Consider over a -form basis for a scalar field . The exterior derivative is: :\begin d\sigma &= du \wedge dx^1 \wedge dx^2 \\ &= \left(\sum_^n \frac \, dx^i\right) \wedge dx^1 \wedge dx^2 \\ &= \sum_^n \left( \frac \, dx^i \wedge dx^1 \wedge dx^2 \right ) \end The last formula follows easily from the properties of the
exterior product In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which are disjoint from S. It is itself an open set and is disjoint from S. The exterior of S in X is often denoted by \operatorname_X S or, if X is ...
. Namely, . Example 2. Let be a -form defined over . By applying the above formula to each term (consider and ) we have the following sum, :\begin d\sigma &= \left( \sum_^2 \frac dx^i \wedge dx \right) + \left( \sum_^2 \frac \, dx^i \wedge dy \right) \\ &= \left(\frac \, dx \wedge dx + \frac \, dy \wedge dx\right) + \left(\frac \, dx \wedge dy + \frac \, dy \wedge dy\right) \\ &= 0 - \frac \, dx \wedge dy + \frac \, dx \wedge dy + 0 \\ &= \left(\frac - \frac\right) \, dx \wedge dy \end


Stokes' theorem on manifolds

If is a compact smooth orientable -dimensional manifold with boundary, and is an -form on , then the generalized form of Stokes' theorem states that: :\int_M d\omega = \int_ \omega Intuitively, if one thinks of 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 .


Further properties


Closed and exact forms

A -form is called ''closed'' if ; closed forms are the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine lea ...
of . is called ''exact'' if for some -form ; exact forms are the
image An SAR radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island. Lava flows a ...
of . Because , every exact form is closed. The
Poincaré lemmaIn mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another diffe ...
states that in a contractible region, the converse is true.


de Rham cohomology

Because the exterior derivative has the property that , it can be used as the differential (coboundary) to define
de Rham cohomology#REDIRECT de Rham cohomology ...
on a manifold. The -th de Rham cohomology (group) is the vector space of closed -forms modulo the exact -forms; as noted in the previous section, the Poincaré lemma states that these vector spaces are trivial for a contractible region, for . For
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an at ...
s, integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology over . 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 is a smooth map and is the contravariant smooth
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps ...
that assigns to each manifold the space of -forms on the manifold, then the following diagram commutes :
none None may refer to: *Zero, the mathematical concept of the quantity "none" *Empty set, the mathematical concept of the collection of things represented by "none" *''none'', an indefinite pronoun in the English language Music *''None'' (Meshuggah EP) ...
so , where denotes the pullback (differential geometry), pullback of . This follows from that , by definition, is , being the Pushforward (differential), pushforward of . Thus is a natural transformation from to .


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 on a real differentiable manifold is a -form. The exterior derivative of this -form is the -form . When an inner product is defined, the gradient of a function is defined as the unique vector in such that its inner product with any element of is the directional derivative of along the vector, that is such that :\langle \nabla f, \cdot \rangle = df = \sum_^n \frac\, dx^i . That is, :\nabla f = (df)^\sharp = \sum_^n \frac\, (dx^i)^\sharp , where denotes the musical isomorphism mentioned earlier that is induced by the inner product. The -form is a section of the cotangent bundle, that gives a local linear approximation to in the cotangent space at each point.


Divergence

A vector field on has a corresponding -form :\begin \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)^v_n \left (dx^1 \wedge \cdots \wedge dx^ \right) \\ &=\sum_^n (-1)^v_i \left (dx^1 \wedge \cdots \wedge dx^ \wedge \widehat \wedge dx^ \wedge \cdots \wedge dx^n \right ) \end where \widehat denotes the omission of that element. (For instance, when , i.e. in three-dimensional space, the -form is locally the scalar triple product with .) The integral of over a hypersurface is the
flux of \mathbf(\mathbf) with the unit normal vector \mathbf(\mathbf) ''(blue arrows)'' at the point \mathbf multiplied by the area dS. The sum of \mathbf\cdot\mathbf dS for each patch on the surface is the flux through the surface Flux describes ...
of over that hypersurface. The exterior derivative of this -form is the -form :d\omega _V = \operatorname V \left (dx^1 \wedge dx^2 \wedge \cdots \wedge dx^n \right ).


Curl

A vector field on also has a corresponding -form :\eta_V = v_1 \, dx^1 + v_2 \, dx^2 + \cdots + v_n \, dx^n. Locally, is the dot product with . The integral of along a path is the Mechanical work, work done against along that path. When , in three-dimensional space, the exterior derivative of the -form is the -form :d\eta_V = \omega_.


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: : \begin \operatorname f &\equiv& \nabla f &=& \left( d f \right)^\sharp \\ \operatorname F &\equiv& \nabla \cdot F &=& \\ \operatorname F &\equiv& \nabla \times F &=& \left( d ( F^\flat ) \right)^\sharp \\ \Delta f &\equiv& \nabla^2 f &=& d d f \\ & & \nabla^2 F &=& \left(dd(F^) - dd(F^)\right)^ , \\ \end where is the Hodge dual, Hodge star operator, and are the musical isomorphisms, is a scalar field and is a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attach ...

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


See also

*Exterior covariant derivative *de Rham complex *Discrete exterior calculus *
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively oriented, ...
*Lie derivative *
Stokes' theorem An illustration of Stokes' theorem, with surface , its boundary and the normal vector . Stokes' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fuji ...
*Fractal derivative


Notes


References

* * * * * * * *


External links

* {{Tensors, state=collapsed Differential forms Differential operators Generalizations of the derivative