Differential form

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

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[edit] 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 dx1, ...,dxn 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 dxI 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.

[edit] 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.

[edit] Formal definition

In differential geometry, a differential form of degree k is a smooth section of the kth 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 Ωk(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".

[edit] Integration of differential 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 differentiable k-manifold over which we wish to integrate, 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

\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

where

\frac{\partial(x^{i_1},\dots,x^{i_k})}{\partial(u^{1},\dots,u^{k})}

is the determinant of the Jacobian. The Jacobian exists because S is differentiable.

See also Stokes' theorem.

[edit] 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.

[edit] 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

\textbf{F} = \frac{1}{2}F_{ab}\, {\mathrm d}x^a \wedge {\mathrm d}x^b.

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

\textbf{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}\, {\textbf{F}} = \textbf{0}
\mathrm{d}\, {*\textbf{F}} = \textbf{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.

[edit] 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.

[edit] See also

[edit] References

  • David Bachman (2006). A Geometric Approach to Differential Forms. Birkhauser. ISBN 978-0-8176-4499-4. 
  • Harley Flanders (1989). Differential forms with applications to the physical sciences. Mineola, NY: Dover Publications. 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.
  • Shigeyuki Morita (2001). Geometry of Differential Forms. AMS. ISBN 0-8218-1045-6. 
  • Walter Rudin (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 0-07-054235-X. 
  • Michael Spivak (1965). Calculus on Manifolds. Menlo Park, CA: W. A. Benjamin. ISBN 0-8053-9021-9. 
  • Vladimir A. Zorich (2004). Mathematical Analysis II. Springer. ISBN 3-540-40633-6.