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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 as being the wedge products of exterior derivatives forming an exterior algebra, was introduced by Élie Cartan.

Contents

Gentle introduction

Properties of the wedge product

Formal definition

Integration of forms

Operations on forms

Differential forms in physics

2-forms in geometric measure theory

See also

References

Gentle introduction

We initially work in an open set in $mathbb\{R\}^n$.
A '0-form' is defined to be a smooth function ''f''.
When we integrate a function ''f'' over an ''m''-dimensional subspace ''S'' of $mathbb\{R\}^n$, we write it as
:$int\_S\; f,\{mathrm\; d\}x^1\; cdots\; \{mathrm\; d\}x^m.$
Consider $\{mathrm\; d\}x^1$, ...,$\{mathrm\; d\}x^n$ for a moment as formal objects themselves, rather than tags appended to make integrals look like Riemann sums.
We call these and their negatives: $-\{mathrm\; d\}x^1,dots,-\{mathrm\; d\}x^n$ ''basic'' 1-''forms''.
We define a "multiplication" rule $wedge$, the wedge product on these elements, making only the ''anticommutativity'' restraint that
:$\{mathrm\; d\}x^i\; wedge\; \{mathrm\; d\}x^j\; =\; -\; \{mathrm\; d\}x^j\; wedge\; \{mathrm\; d\}x^i$
for all ''i'' and ''j''. Note that this implies
:$\{mathrm\; d\}x^i\; wedge\; \{mathrm\; d\}x^i\; =\; 0$.
We define the set of all these products to be ''basic'' 2-''forms'', and similarly we define the set of products
:$\{mathrm\; d\}x^i\; wedge\; \{mathrm\; d\}x^j\; wedge\; \{mathrm\; d\}x^k$
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
:$(f,\{mathrm\; d\}x^I\; +\; g,\{mathrm\; d\}x^J)wedge(p,\{mathrm\; d\}x^K\; +\; q,\{mathrm\; d\}x^L)\; =$
::$f\; cdot\; p,\{mathrm\; d\}x^I\; wedge\; \{mathrm\; d\}x^K\; +$
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,

etc., where $\{mathrm\; d\}x^I$ 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 manifolds. To this end, suppose we have an open coordinate 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
:$w\; wedge\; (f\; +\; g)\; =\; w\; wedge\; f\; +\; w\; wedge\; g.$
Also, if ''f'' is a ''k''-form and ''g'' is an ''l''-form, then:
:$f\; wedge\; g\; =\; (-1)^\{kl\}\; g\; wedge\; f.$

Formal definition

In differential geometry, a 'differential form' of degree ''k'' is a smooth section of the ''k''th 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^k(M)''.
''k''-forms can be defined as totally antisymmetric covariant tensor fields.
For example, the differential of a smooth function on a manifold (a 0-form) is a 1-form.
1-forms are a particularly useful basic concept in the coordinate-free treatment of tensors. 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 "covariant vector fields", "covector fields", or "dual vector fields".

Integration of forms

Differential forms of degree ''k'' are integrated over ''k'' dimensional chains. 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
:$omega=sum\; a\_\{i\_1,dots,i\_k\}(\{mathbf\; x\}),\{mathrm\; d\}x^\{i\_1\}\; wedge\; cdots\; wedge\; \{mathrm\; d\}x^\{i\_k\}$
be a differential form and ''S'' a set for which we wish to integrate over, where ''S'' has the parameterization
:$S(\{mathbf\; u\})=(x^1(\{mathbf\; u\}),dots,x^n(\{mathbf\; u\}))$
for 'u' in the parameter domain ''D''. Then [Rudin, 1976] defines the integral of the differential form over ''S'' as
:
where
:
is the determinant of the Jacobian.
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 dual, codifferential and Lie derivative. One important property of the exterior derivative is that d² = 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 of chains.

Differential forms in physics

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
:
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
:$extbf\{J\}\; =\; J^a\; epsilon\_\{abcd\},\; \{mathrm\; d\}x^b\; wedge\; \{mathrm\; d\}x^c\; wedge\; \{mathrm\; d\}x^d$
Using these definitions, Maxwell's equations can be written very compactly in geometrized units as
:$mathrm\{d\},\; \{$
★ extbf{F}} = extbf{J}
where $$
★ denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general.
The 2-form $$
★ mathbf{F} 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 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)

References

★ A Geometric Approach to Differential Forms, David Bachman, , , Birkhauser, 2006, ISBN 978-0-8176-4499-4

★ Differential forms with applications to the physical sciences, Harley Flanders, , , Dover Publications, 1989, ISBN 0-486-66169-5

★ Principles of Mathematical Analysis, Walter Rudin, , , McGraw-Hill, 1976, ISBN 0-07-054235-X

★ Calculus on Manifolds, Michael Spivak, , , W. A. Benjamin, 1965, ISBN 0-8053-9021-9

★ Mathematical Analysis II, Vladimir A. Zorich, , , Springer, 2004, ISBN 3-540-40633-6