View source for Exterior derivative

{{Short description|operation of differentiation in differential geometry}}
{{Calculus |Multivariable}}
{{no footnotes|date=July 2019}}


On a [[differentiable manifold]], the '''exterior derivative''' extends the concept of the [[pushforward (differential)|differential]] of a function to [[differential form]]s of higher degree. The exterior derivative was first described in its current form by [[Élie Cartan]] in 1899; it allows for a natural, metric-independent generalization of [[Stokes' theorem]], [[Gauss's theorem]], and [[Green's theorem]] from vector calculus.

If a {{math|''k''}}-form is thought of as measuring the flux through an infinitesimal {{math|''k''}}-[[Parallelepiped#Parallelotope|parallelotope]], then its exterior derivative can be thought of as measuring the net flux through the boundary of a {{math|(''k'' + 1)}}-parallelotope.

== Definition ==
The exterior derivative of a [[differential form]] of degree {{math|''k''}} is a differential form of degree {{math|''k'' + 1.}}

If {{math|&thinsp;''f''&thinsp;}} is a [[Smoothness|smooth function]] (a {{math|0}}-form), then the exterior derivative of {{math|&thinsp;''f''&thinsp;}} is the [[Pushforward (differential)|differential]] of {{math|&thinsp;''f''&thinsp;}}. That is, {{math|''df''&thinsp;}} is the unique [[1-form|{{math|1}}-form]] such that for every smooth [[vector field]] {{math|''X''}}, {{math|1=''df''&thinsp;(''X'') = ''d''<sub>''X''</sub>&thinsp;''f''&thinsp;}}, where {{math|''d''<sub>''X''</sub>&thinsp;''f''&thinsp;}} is the [[directional derivative]] of {{math|&thinsp;''f''&thinsp;}} in the direction of {{math|''X''}}.

There are a variety of equivalent definitions of the exterior derivative of a general {{math|''k''}}-form.

===In terms of axioms===
The exterior derivative is defined to be the unique {{math|ℝ}}-linear mapping from {{math|''k''}}-forms to {{math|(''k'' + 1)}}-forms satisfying the following properties:

# {{math|''df''&thinsp;}} is the [[Differential_of_a_function|differential]] of {{math|&thinsp;''f''&thinsp;}}, for {{math|0}}-forms  ([[Smoothness|smooth functions]]) {{math|&thinsp;''f''&thinsp;}}.
# {{math|1=''d''(''df''&thinsp;) = 0}} for any {{math|0}}-form (smooth function) {{math|&thinsp;''f''&thinsp;}}.
# {{math|1=''d''(''α'' ∧ ''β'') = ''dα'' ∧ ''β'' + (−1){{sup|''p''}} (''α'' ∧ ''dβ'')}} where {{mvar|α}} is a {{math|''p''}}-form. That is to say, {{math|''d''}} is an [[derivation (algebra)|antiderivation]] of degree {{math|1}} on the [[exterior algebra]] of differential forms.

The second defining property holds in more generality: in fact, {{math|1=''d''(''dα'') = 0}} for any {{math|''k''}}-form {{mvar|α}}; more succinctly, {{math|1=''d''{{i sup|2}} = 0}}. The third defining property implies as a special case that if {{math|&thinsp;''f''&thinsp;}} is a function and {{mvar|α}} a {{math|''k''}}-form, then {{math|1=''d''(&thinsp;''fα'') = ''d''(&thinsp;''f'' ∧ ''α'') = ''df''&thinsp; ∧ ''α'' + &thinsp;''f''&thinsp; ∧ ''dα''}} because functions are {{math|0}}-forms, 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]] {{math|(''x''{{sup|1}}, ..., ''x''{{i sup|''n''}})}}. The coordinate differentials {{math|''dx''{{sup|1}}, ..., ''dx''{{i sup|''n''}}}} form a basis of the space of one-forms, each associated with a coordinate. Given a [[multi-index]] {{math|1=''I'' = (''i''{{sub|1}}, ..., ''i''{{sub|''k''}})}} with {{math|1 ≤ ''i''{{sub|''p''}} ≤ ''n''}} for {{math|1 ≤ ''p'' ≤ ''k''}} (and denoting {{math|''dx''{{i sup|''i''{{sub|1}}}} ∧ ... ∧ ''dx''{{i sup|''i''{{sub|''k''}}}}}} with an [[abuse of notation]] {{math|1=''dx''{{i sup|''I''}}}}), the exterior derivative of a (simple) {{math|''k''}}-form

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

over {{math|ℝ{{sup|''n''}}}} is defined as

:<math>d{\varphi} =  \frac{\partial g}{\partial x^i} dx^i \wedge dx^I</math>

(using [[Einstein notation]]).  The definition of the exterior derivative is extended [[linear]]ly to a general {{math|''k''}}-form

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

where each of the components of the multi-index {{math|''I''}} run over all the values in {{math|{1, ..., ''n''}<nowiki/>}}. Note that whenever {{math|''i''}} equals one of the components of the multi-index {{math|''I''}} then {{math|1=''dx''{{i sup|''i''}} ∧ ''dx''{{i sup|''I''}} = 0}} (see [[Exterior product]]).

The definition of the exterior derivative in local coordinates follows from the preceding [[#In terms of axioms|definition in terms of axioms]]. Indeed, with the {{math|''k''}}-form {{math|''φ''}} 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 d
                      x^{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 {{math|''g''}} as a {{math|0}}-form, and then applied the properties of the exterior derivative.

This result extends directly to the general {{math|''k''}}-form {{math|''ω''}} as

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

In particular, for a {{math|1}}-form {{math|''ω''}}, the components of {{math|''dω''}} in [[Local coordinate system|local coordinates]] are
:<math>(d\omega)_{ij} = \partial_i \omega_j - \partial_j \omega_i. </math>

===In terms of invariant formula===
Alternatively, an explicit formula can be given for the exterior derivative of a {{math|''k''}}-form {{math|''ω''}}, when paired with {{math|''k'' + 1}} arbitrary smooth [[vector field]]s {{math|''V''<sub>0</sub>,''V''<sub>1</sub>, ..., ''V''<sub>''k''</sub>}}:

:<math>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 )</math>

where {{math|[''V<sub>i</sub>'', ''V<sub>j</sub>'']}} denotes the [[Lie bracket of vector fields|Lie bracket]] and a hat denotes the omission of that element:

:<math>\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 ).</math>

In particular, for {{math|1}}-forms we have: {{math|1=''dω''(''X'', ''Y'') = ''X''(''ω''(''Y'')) − ''Y''(''ω''(''X'')) − ''ω''([''X'', ''Y''])}}, where {{math|''X''}} and {{math|''Y''}} are vector fields, {{math|''X''(''ω''(''Y''))}} is the scalar field defined by the vector field {{math|''X'' ∈ Γ(''TM'')}} applied as a differential operator ("directional derivative along ''X''") to the scalar field defined by applying {{math|''ω'' ∈ Γ{{sup|∗}}(''TM'')}} as a covector field to the vector field {{math|''Y'' ∈ Γ(''TM'')}} and likewise for {{math|''Y''(''ω''(''X''))}}.

'''Note:''' Some authors (e.g., Kobayashi–Nomizu and Helgason) use a formula that differs by a factor of {{math|{{sfrac|''k'' + 1}}}}:
:<math>\begin{align}d\omega(V_0,...,V_k) &= {1 \over k+1} \sum_i(-1)^{i} V_i \left( \omega \left (V_0, \ldots, \hat V_i, \ldots,V_k \right )\right) \\
&+ {1 \over k+1}  \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 ).\end{align}</math>

== Examples ==
'''Example 1.''' Consider {{math|1=''σ'' = ''u''&thinsp;''dx''{{i sup|1}} ∧ ''dx''{{i sup|2}}}} over a {{math|1}}-form basis {{math|''dx''{{i sup|1}}, ..., ''dx''{{i sup|''n''}}}} for a scalar field {{math|''u''}}. 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 follows easily from the properties of the [[exterior product]]. Namely, {{math|1=''dx''{{i sup|''i''}} ∧ ''dx''{{i sup|''i''}} = 0}}.

'''Example 2.''' Let {{math|1=''σ'' = ''u''&thinsp;''dx'' + ''v''&thinsp;''dy''}} be a {{math|1}}-form defined over {{math|ℝ{{sup|2}}}}. By applying the above formula to each term (consider {{math|1=''x''{{i sup|1}} = ''x''}} and {{math|1=''x''{{i sup|2}} = ''y''}}) we have the following 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 ==
{{main|Stokes' theorem}}

If {{math|''M''}} is a compact smooth orientable {{math|''n''}}-dimensional manifold with boundary, and {{math|''ω''}} is an {{math|(''n'' − 1)}}-form on {{math|''M''}}, then the generalized form of [[Stokes' theorem]] states that:

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

Intuitively, if one thinks of {{math|''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 {{math|''M''}}.

== Further properties ==

===Closed and exact forms===
{{main article|Closed and exact forms}}
A {{math|''k''}}-form {{math|''ω''}} is called ''closed'' if {{math|1=''dω'' = 0}}; closed forms are the [[Kernel (algebra)|kernel]] of {{math|''d''}}. {{math|''ω''}} is called ''exact'' if {{math|1=''ω'' = ''dα''}} for some {{math|(''k'' − 1)}}-form {{math|''α''}}; exact forms are the [[Image (mathematics)|image]] of {{math|''d''}}. Because {{math|1=''d''{{i sup|2}} = 0}}, 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 {{math|''d''}} has the property that {{math|1=''d''{{i sup|2}} = 0}}, it can be used as the [[Cochain complex|differential]] (coboundary) to define [[de Rham cohomology]] on a manifold. The {{math|''k''}}-th de Rham cohomology (group) is the vector space of closed {{math|''k''}}-forms modulo the exact {{math|''k''}}-forms; as noted in the previous section, the Poincaré lemma states that these vector spaces are trivial for a contractible region, for {{math|''k'' > 0}}. For [[smooth manifold]]s, integration of forms gives a natural homomorphism from the de Rham cohomology to the singular cohomology over {{math|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 [[Chain complex#Formal definition|boundary map]] on singular simplices.

===Naturality===
The exterior derivative is natural in the technical sense: if {{math|&thinsp;''f'' : ''M'' → ''N''}} is a smooth map and {{math|Ω{{sup|''k''}}}} is the contravariant smooth [[functor]] that assigns to each manifold the space of {{math|''k''}}-forms on the manifold, then the following diagram commutes

:[[Image:Exteriorderivnatural.png|none]]

so {{math|1=''d''(&thinsp;''f''{{i sup|∗}}''ω'') = &thinsp;''f''{{i sup|∗}}''dω''}}, where {{math|&thinsp;''f''{{i sup|∗}}}} denotes the [[pullback (differential geometry)|pullback]] of {{math|&thinsp;''f''&thinsp;}}. This follows from that {{math|&thinsp;''f''{{i sup|∗}}''ω''(·)}}, by definition, is {{math|''ω''(&thinsp;''f''<sub>∗</sub>(·))}}, {{math|&thinsp;''f''<sub>∗</sub>}} being the [[Pushforward (differential)|pushforward]] of {{math|&thinsp;''f''&thinsp;}}. Thus {{math|''d''}} is a [[natural transformation]] from {{math|Ω{{sup|''k''}}}} to {{math|Ω{{sup|''k''+1}}}}.

== 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]] {{math|&thinsp;''f'' : ''M'' → ℝ}} on a real differentiable manifold {{math|''M''}} is a {{math|0}}-form. The exterior derivative of this {{math|0}}-form is the {{math|1}}-form {{math|''df''}}.

When an inner product {{math|{{langle}}·,·{{rangle}}}} is defined, the [[gradient]] {{math|∇''f''&thinsp;}} of a function {{math|&thinsp;''f''&thinsp;}} is defined as the unique vector in {{math|''V''}} such that its inner product with any element of {{math|''V''}} is the directional derivative of {{math|&thinsp;''f''&thinsp;}} 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}\, (dx^i)^\sharp ,</math>
where {{math|{{music|sharp}}}} denotes the [[musical isomorphism]] {{math|{{music|sharp}} : ''V''{{sup|∗}} → ''V''}} mentioned earlier that is induced by the inner product.

The {{math|1}}-form {{math|''df''&thinsp;}} is a section of the [[cotangent bundle]], that gives a local linear approximation to {{math|&thinsp;''f''&thinsp;}} in the cotangent space at each point.

===Divergence===
A vector field {{math|1=''V'' = (''v''<sub>1</sub>, ''v''<sub>2</sub>, ... ''v<sub>n</sub>'')}} on {{math|ℝ{{sup|''n''}}}} has a corresponding {{math|(''n'' − 1)}}-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 {{math|1=''n'' = 3}}, i.e. in three-dimensional space, the {{math|2}}-form {{math|''ω<sub>V</sub>''}} is locally the [[scalar triple product]] with {{math|''V''}}.)  The integral of {{math|''ω<sub>V</sub>''}} over a hypersurface is the [[flux]] of {{math|''V''}} over that hypersurface.

The exterior derivative of this {{math|(''n'' − 1)}}-form is the {{math|''n''}}-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 {{math|''V''}} on {{math|ℝ{{sup|''n''}}}} also has a corresponding {{math|1}}-form

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

Locally, {{math|''η<sub>V</sub>''}} is the dot product with {{math|''V''}}. The integral of {{math|''η<sub>V</sub>''}} along a path is the [[Mechanical work|work]] done against {{math|−''V''}} along that path.

When {{math|1=''n'' = 3}}, in three-dimensional space, the exterior derivative of the {{math|1}}-form {{math|''η<sub>V</sub>''}} is the {{math|2}}-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} ( F^\flat )} \\
      \operatorname{curl} F &\equiv& \nabla \times F &=& \left( {\star} d ( F^\flat ) \right)^\sharp \\
      \Delta f              &\equiv& \nabla^2 f      &=& {\star} d {\star} d f \\
                            &      & \nabla^2 F      &=& \left(d{\star}d{\star}(F^{\flat}) - {\star}d{\star}d(F^{\flat})\right)^{\sharp} , \\
\end{array}
</math>

where {{math|⋆}} is the [[Hodge dual|Hodge star operator]], {{music|flat}} and {{music|sharp}} are the [[musical isomorphism]]s, {{mvar|&thinsp;f&thinsp;}} is a [[scalar field]] and {{mvar|F}} is a [[vector field]].

Note that the expression for <math>\operatorname{curl}</math> makes sense only in three dimensions, since it requires <math>\sharp</math> to act on <math>{\star} d( F^\flat )</math>, which is a form of degree <math>n-2</math>.

== See also ==
*[[Exterior covariant derivative]]
*[[de Rham complex]]
*[[Discrete exterior calculus]]
*[[Green's theorem]]
*[[Lie derivative]]
*[[Stokes' theorem]]
*[[Fractal derivative]]

== Notes ==
<!--- See [[Wikipedia:Footnotes]] on how to create references using <ref></ref> tags which will then appear here automatically -->
{{Reflist|30em}}

== References ==
* {{cite journal | last =Cartan | first =Élie | author-link =Élie Cartan
 | title =Sur certaines expressions différentielles et le problème de Pfaff
 | journal =Annales Scientifiques de l'École Normale Supérieure |series=Série 3
 | volume =16 | pages =239–332 | publisher =Gauthier-Villars | location =Paris | date =1899 | language =French
 | url =http://www.numdam.org/item?id=ASENS_1899_3_16__239_0
 | issn =0012-9593 | jfm =30.0313.04 | access-date = 2 Feb 2016}}
* {{cite book |author=Conlon, Lawrence |title=Differentiable manifolds |publisher=Birkhäuser |location=Basel, Switzerland |year=2001 |pages= 239 |isbn=0-8176-4134-3 |oclc= |doi=}}
* {{cite book |author=Darling, R. W. R. |title=Differential forms and connections |publisher=Cambridge University Press |location=Cambridge, UK |year=1994 |pages=35 |isbn=0-521-46800-0 |oclc= |doi=}}
* {{cite book |author=Flanders, Harley |title=Differential forms with applications to the physical sciences |publisher=Dover Publications |location=New York |year=1989 |pages=20 |isbn=0-486-66169-5 |oclc= |doi=}}
* {{cite book|url=https://archive.org/details/LoomisL.H.SternbergS.AdvancedCalculusRevisedEditionJonesAndBartlett|title=Advanced Calculus|first=Lynn H.|last2=Sternberg|first2=Shlomo|publisher=Jones and Bartlett|year=1989|isbn=0-486-66169-5|location=Boston|pages=304–473 (ch. 7–11)|chapter=|doi=|oclc=|author=Loomis}}
* {{cite book |author=Ramanan, S. |title=Global calculus |publisher=American Mathematical Society |location=Providence, Rhode Island |year=2005 |pages=54 |isbn=0-8218-3702-8 |oclc= |doi=}}
* {{cite book | last =Spivak | first =Michael | author-link =Michael Spivak | title =[[Calculus on Manifolds (book)|Calculus on Manifolds]]
 | publisher =Westview Press | date =1971 | location =Boulder, Colorado | url = | doi = | isbn =9780805390216 }}
* {{citation|last=Warner|first= Frank W.|title= Foundations of differentiable manifolds and Lie groups|series= Graduate Texts in Mathematics|volume= 94|publisher= Springer|year=1983|isbn= 0-387-90894-3}}

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[[Category:Differential forms]]
[[Category:Differential operators]]
[[Category:Generalizations of the derivative]]