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

1 Gentle introduction
2 Properties of the wedge product
3 Formal definition
4 Integration of forms
5 Operations on forms
6 Differential forms in physics
7 See also
8 References

[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 $d x 1$ , ..., $d x n$ for a moment as formal objects themselves, rather than tags appended to make integrals look like Riemann sums. We call these and their negatives: $- d x 1,..., - d x n$ basic 1-forms.

We define a "multiplication" rule ∧, 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 $d x I$ 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 what that means, see manifold.

[edit] Properties of the wedge product

It can be proved 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 k-form is easily remembered by noting that it is a total antisymmetric covariant tensor.

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 can be defined as real-valued linear functions of vectors, and they can be seen to create a dual space with regard to the vector space of the vectors they are defined over. An older name for 1-forms in this context is "covariant vectors". Other names for them are "covector" and "dual vector".

[edit] 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

$\int_S \omega =\int_D \sum a_{i_1,\dots,i_k}(S({\mathbf u})) \frac{\partial(x^{i_1},\dots,x^{i_k})}{\partial(u^{1},\dots,u^{k})}\,\mathrm{d}{\mathbf u}$

where

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

is the determinant of the Jacobian.

[edit] Operations on forms

The set of all k-forms on a manifold is a vector space commonly denoted Ω^k(M). Furthermore, there are several important operations one can perform on a differential form: wedge product, exterior derivative (denoted by d), 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.

One simple and important operation is contraction by a vector field V, which is a linear map $i$ _V: Ω^k(M) → Ω^k − 1(M). When ω = ω(v₁, ..., v_k) is a k-form, one merely inserts V into the first slot, resulting in a (k − 1)-form $i$ _Vω = ω(V, v₂, ..., v_k).

[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

$\mathrm{d} * \textbf{F} = \textbf{J}$

where $*$ denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general.

2-Form $* \mathbf{F}$ is also called Maxwell 2-form.

[edit] See also

complex differential form

[edit] References

Harley Flanders (1989). Differential forms with applications to the physical sciences. Mineola, NY: Dover Publications. ISBN 0-486-66169-5.
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.