Differential form
221519
211181766
2008-05-09T02:49:27Z
Oleg Alexandrov
153314
Reverted edits by [[Special:Contributions/128.36.157.183|128.36.157.183]] ([[User talk:128.36.157.183|talk]]) to last version by 24.17.31.87
A '''differential form''' is a mathematical concept in the fields of [[multivariate calculus]], [[differential topology]] and [[tensors]]. The modern notation for the differential form, as well as the idea of the differential forms being the [[wedge product]]s of [[exterior derivative]]s forming an [[exterior algebra]], was introduced by [[Élie Cartan]].
==Gentle introduction==
We initially work in an [[open set]] in <math>\mathbb{R}^n</math>.
A '''0-form''' is defined to be a [[smooth function]] ''f''.
When we [[integral|integrate]] a [[function (mathematics)|function]] ''f'' over an ''m''-[[dimension]]al subspace ''S'' of <math>\mathbb{R}^n</math>, we write it as
:<math>\int_S f\,{\mathrm d}x^1 \cdots {\mathrm d}x^m.</math>
Consider <math>{\mathrm d}x^1</math>, ...,<math>{\mathrm d}x^n</math> for a moment as formal objects themselves, rather than tags appended to make integrals look like [[Riemann sum]]s.
We call these and their negatives: <math>-{\mathrm d}x^1,\dots,-{\mathrm d}x^n</math> ''basic'' [[one-form|1-''forms'']].
We define a "multiplication" rule <math>\wedge</math>, the [[wedge product]] on these elements, making only the ''[[anticommutativity]]'' restraint that
:<math>{\mathrm d}x^i \wedge {\mathrm d}x^j = - {\mathrm d}x^j \wedge {\mathrm d}x^i</math>
for all ''i'' and ''j''. Note that this implies
:<math>{\mathrm d}x^i \wedge {\mathrm d}x^i = 0</math>.
We define the set of all these products to be ''basic'' 2-''forms'', and similarly we define the set of products
:<math>{\mathrm d}x^i \wedge {\mathrm d}x^j \wedge {\mathrm d}x^k</math>
to be ''basic'' 3-''forms'', assuming ''n'' is at least 3. Now define a ''monomial k''-''form'' to be a 0-form times a basic ''k''-form for all ''k'', and finally define a '''''k''-form''' to be a sum of monomial ''k''-forms.
We extend the wedge product to these sums by defining
:<math>(f\,{\mathrm d}x^I + g\,{\mathrm d}x^J)\wedge(p\,{\mathrm d}x^K + q\,{\mathrm d}x^L) = </math>
::<math>f \cdot p\,{\mathrm d}x^I \wedge {\mathrm d}x^K +
f \cdot q\,{\mathrm d}x^I \wedge {\mathrm d}x^L +
g \cdot p\,{\mathrm d}x^J \wedge {\mathrm d}x^K +
g \cdot q\,{\mathrm d}x^J \wedge {\mathrm d}x^L,
</math>
etc., where <math>{\mathrm d}x^I</math> and friends represent basic ''k''-forms. In other words, the product of sums is the sum of all possible products.
Now, we also want to define ''k''-forms on smooth [[manifold]]s. To this end, suppose we have an open coordinate [[cover (topology)|cover]]. We can define a ''k''-form on each coordinate neighborhood; a '''global ''k''-form''' is then a set of ''k''-forms on the coordinate neighborhoods such that they agree on the overlaps. For a more precise definition of what that means, see [[manifold]].
==Properties of the wedge product==
It can be proven that if ''f'', ''g'', and ''w'' are any differential forms, then
:<math>w \wedge (f + g) = w \wedge f + w \wedge g. </math>
Also, if ''f'' is a ''k''-form and ''g'' is an ''l''-form, then:
:<math>f \wedge g = (-1)^{kl} g \wedge f.</math>
==Formal definition==
In [[differential geometry]], a '''differential form''' of degree ''k'' is a smooth [[Section (fiber bundle)|section]] of the ''k''th [[wedge product|exterior power]] of the [[cotangent bundle]] of a [[manifold]]. At any point ''p'' on a manifold, a ''k''-form gives a [[multilinear map]] from the ''k''-th exterior power of the [[tangent space]] at ''p'' to '''R'''.
The set of all ''k''-forms on a manifold ''M'' is a [[vector space]] commonly denoted ''Ω<sup>k</sup>(M)''.
''k''-forms can be defined as totally [[antisymmetric]] [[covariant]] [[tensor]] fields.
For example, the [[differential (calculus)|differential]] of a smooth function on a manifold (a 0-form) is a [[one-form|1-form]].
[[one-form|1-forms]] are a particularly useful basic concept in the coordinate-free treatment of [[tensor]]s. In this context, they assign, to each point of a manifold, a [[linear functional]] on the tangent space at that point. In this setting, particularly in the physics literature, 1-forms are sometimes called "[[covariance and contravariance|covariant]] vector fields", "covector fields", or "dual vector fields".
==Integration of differential forms==
Differential forms of degree ''k'' are integrated over ''k'' dimensional [[chain (algebraic topology)|chain]]s. If ''k'' = 0, this is just evaluation of functions at points. Other values of ''k'' = 1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals etc.
Let
:<math>\omega=\sum a_{i_1,\dots,i_k}({\mathbf x})\,{\mathrm d}x^{i_1} \wedge \cdots \wedge {\mathrm d}x^{i_k} </math>
be a differential form and ''S'' a [[differentiable manifold|differentiable k-manifold]] over which we wish to integrate, where ''S'' has the parameterization
:<math>S({\mathbf u})=(x^1({\mathbf u}),\dots,x^n({\mathbf u}))</math>
for '''u''' in the parameter domain ''D''. Then [Rudin, 1976] defines the integral of the differential form over ''S'' as
:<math>\int_S \omega =\int_D \sum a_{i_1,\dots,i_k}(S({\mathbf u})) \frac{\partial(x^{i_1},\dots,x^{i_k})}{\partial(u^{1},\dots,u^{k})}\,du^1\ldots du^k</math>
where
:<math>\frac{\partial(x^{i_1},\dots,x^{i_k})}{\partial(u^{1},\dots,u^{k})}</math>
is the determinant of the [[Jacobian]].
The Jacobian exists because ''S'' is differentiable.
See also [[Stokes' theorem]].
==Operations on forms==
There are several important operations one can perform on a differential form: [[wedge product]], [[exterior derivative]] (denoted by d), [[interior product]], [[Hodge star|Hodge dual]], [[Hodge dual#The codifferential|codifferential]] and [[Lie derivative]]. One important property of the exterior derivative is that d<sup>2</sup> = 0; see [[de Rham cohomology]] for more details.
The fundamental relationship between the exterior derivative and integration
is given by the general [[Stokes' theorem]], which also provides the duality between [[de Rham cohomology]] and the [[homology (mathematics)|homology]] of chains.
==Differential forms in physics==<!-- This section is linked from [[Maxwell's equations]] -->
Differential forms arise in some important physical contexts. For example, in Maxwell's theory of [[electromagnetism]], the '''Faraday 2-form''' or [[electromagnetic field strength]] is
:<math>\textbf{F} = \frac{1}{2}F_{ab}\, {\mathrm d}x^a \wedge {\mathrm d}x^b.</math>
Note that this form is a special case of the [[curvature form]] on the [[U(1)]] [[principal fiber bundle]] on which both electromagnetism and general [[gauge theories]] may be described. The '''''current 3-form''''' is
:<math>\textbf{J} = J^a \epsilon_{abcd}\, {\mathrm d}x^b \wedge {\mathrm d}x^c \wedge {\mathrm d}x^d.</math>
Using these definitions, [[Maxwell's equations]] can be written very compactly in [[geometrized units]] as
:<math>\mathrm{d}\, {\textbf{F}} = \textbf{0}</math>
:<math>\mathrm{d}\, {*\textbf{F}} = \textbf{J}</math>
where <math>*</math> denotes the [[Hodge star]] operator. Similar considerations describe the geometry of gauge theories in general.
The 2-form <math>* \mathbf{F}</math> is also called '''Maxwell 2-form'''.
==2-forms in geometric measure theory==
Numerous minimality results for complex analytic manifolds are based on the [[Wirtinger inequality (2-forms)|Wirtinger inequality for 2-forms]]. A succinct proof may be found in [[Herbert Federer]]'s classic text Geometric Measure Theory. The Wirtinger inequality is also a key ingredient in [[Gromov's inequality for complex projective space]] in [[systolic geometry]].
==See also==
* [[complex differential form]]
* [[vector-valued differential form]]
* [[Wirtinger inequality (2-forms)]]
* http://www.sjsu.edu/faculty/watkins/difforms.htm
==References==
* {{cite book | author=David Bachman | title=A Geometric Approach to Differential Forms | publisher=Birkhauser | year = 2006 |id=ISBN 978-0-8176-4499-4 }}
* {{cite book | author=Harley Flanders | title=Differential forms with applications to the physical sciences | location=Mineola, NY | publisher=Dover Publications | year=1989 | id=ISBN 0-486-66169-5}}
* Wendell H. Fleming (1965) ''Functions of Several Variables'', Addison-Wesley. Chapter 6: Exterior algebra and differential calculus, pages 205-38. This textbook in [[multivariate calculus]] introduces the exterior algebra of differential forms adroitly into the calculus sequence for colleges.
* {{cite book | author=Shigeyuki Morita | title=Geometry of Differential Forms | publisher=AMS | year=2001 | id=ISBN 0-8218-1045-6}}
* {{cite book | author=[[Walter Rudin]] | title=Principles of Mathematical Analysis | location=New York | publisher=McGraw-Hill | year=1976 | id=ISBN 0-07-054235-X}}
* {{cite book | author=[[Michael Spivak]] | title=Calculus on Manifolds | location=Menlo Park, CA | publisher=W. A. Benjamin | year = 1965 |id=ISBN 0-8053-9021-9}}
* {{cite book | author=Vladimir A. Zorich | title=Mathematical Analysis II | publisher=Springer | year = 2004 |id=ISBN 3-540-40633-6 }}
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