- Exterior derivative
In

differential geometry , the**exterior derivative**extends the concept of the differential of a function, which is a form of degree zero, todifferential form s of higher degree. Its current form was invented byÉlie Cartan .The exterior derivative "d" has the property that "d"

^{2}= 0 and is thedifferential (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 asmooth 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.**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 $1le\; i\_1,\; i\_2,\; dots,\; i\_k\; le\; n,$ the exterior derivative of a "k"-form: $omega\; =\; f\_Idx\_I=f\_\{i\_1i\_2cdots\; i\_k\}dx\_\{i\_1\}wedge\; dx\_\{i\_2\}wedgecdotswedge\; dx\_\{i\_k\}$

over

**R**^{"n"}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 extendedlinear ly. Note that whenever $i$ is one of the components of the multi-index $I,$ then $dx\_i\; wedge\; dx\_I\; =\; 0$ (seewedge product ).Geometrically, the "k" + 1 form "d"ω acts on each

tangent space of**R**^{"n"}in the following way: a ("k" + 1)-tuple of vectors ("u"_{1},...,"u"_{"k" + 1}) in the tangent space defines an oriented ("k" + 1)-polyhedron "p". "d"ω("u"_{1},...,"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 immediatelyStokes' theorem .**Examples**For a 1-form $sigma\; =\; u,\; dx\; +\; v,\; dy$ on

**R**^{"2"}we have, by applying the above formula to each term,:$d\; sigma\; =\; left(frac\{partial\{u\{partial\{x\; dx\; wedge\; dx\; +\; frac\{partial\{u\{partial\{y\; dy\; wedge\; dx\; ight)\; +\; left(frac\{partial\{v\{partial\{x\; dx\; wedge\; dy\; +\; frac\{partial\{v\{partial\{y\; dy\; wedge\; dy\; ight)$::$=\; 0\; -frac\{partial\{u\{partial\{y\; dx\; wedge\; dy\; +\; frac\{partial\{v\{partial\{x\; dx\; wedge\; dy\; +\; 0$:: $=\; left(frac\{partial\{v\{partial\{x\; -\; frac\{partial\{u\{partial\{y\; ight)\; dx\; wedge\; dy.$

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

*the

wedge product rule holds (see antiderivation)::$d(omega\; wedge\; eta)\; =\; domega\; wedge\; eta+(-1)^$ m deg,}omega}(omega wedge deta),

* and "d"

^{2}= 0, which follows from the equality ofmixed 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 differential s"). 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 smoothfunctor that assigns to each manifold the space of "k"-forms on the manifold, then the following diagram commutesso "d"("f"*ω) = "f"*"d"ω, 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 anatural transformation from Ω^{"k"}to Ω^{"k"+1}.**Invariant formula**Given a "k"-form "ω" and arbitrary smooth

vector field s "V_{0},V_{1}, …, V_{k}" we have:$domega(V\_0,V\_1,...V\_k)\; =\; sum\_i(-1)^i\; V\_ileft(omega(V\_0,\; ldots,\; hat\; V\_i,\; ldots,V\_k)\; ight)$

::$+sum\_\{i\}(-1)^\{i+j\}omega(\; [v\_i,\; v\_j]\; ,\; v\_0,\; ldots,\; hat\; v\_i,\; v\_j,\; v\_k)\; math>$

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::$d\; omega(X,Y)\; =\; X(omega(Y))\; -\; Y(omega(X))\; -\; omega(\; [X,Y]\; ).$

**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.**Gradient**For a 0-form, that is, a

smooth function "f":**R**^{"n"}→**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", :$df(V)\; =\; langle\; mbox\{grad\}f\; ,V\; angle,$

where grad "f" denotes

gradient of "f" and < , > is thescalar product .**Curl**One can associate to a vector field "V" = ("u", "v", "w") on

**R**^{3}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 ω

^{1}_{"V"}over a path gives work done against -"V" along the path; locally, it is the dot product with "V". The integral of ω^{2}_{"V"}over a surface gives the flux of "V" over that surface; locally, it is thescalar 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 ω^{1}_{"V"}over the boundary of the surface.**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".**Invariant formulations of div, grad, and curl**The three operators above can be written in coordinate-free notation as follows:

: $egin\{array\}\{rcl\}\; abla\; f\; =\; left(\; \{mathbf\; d\}\; f\; ight)^sharp\; \backslash \; abla\; imes\; F\; =\; left\; [\; star\; left(\; \{mathbf\; d\}\; F^flat\; ight)\; ight]\; ^sharp\; \backslash \; abla\; cdot\; F\; =\; star\; \{mathbf\; d\}\; left(\; star\; F^flat\; ight)\; \backslash end\{array\}$

where $star$ is the Hodge star operator and $flat$ and $sharp$ are the musical isomorphisms.

**ee also***Exterior covariant derivative

*Green's theorem

*Lie derivative

*Discrete exterior calculus **References***

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