In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to defining integrands over curves, surfaces, volumes, and higher dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression f(x) dx from onevariable calculus is called a 1form, and can be integrated over an interval [a,b] in the domain of f:
 $\backslash int\_a^b\; f(x)\backslash ,dx$
and similarly the expression: f(x,y,z) dx∧dy + g(x,y,z) dx∧dz + h(x,y,z) dy∧dz is a 2form
that has a surface integral over an oriented surface S:
 $\backslash int\_S\; f(x,y,z)\backslash ,dx\backslash wedge\; dy\; +\; g(x,y,z)\backslash ,dx\backslash wedge\; dz\; +\; h(x,y,z)\backslash ,dy\backslash wedge\; dz.$
Likewise, a 3form f(x, y, z) dx∧dy∧dz represents a volume element that can be integrated over a region of space.
The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration. There is an operation d on differential forms known as the exterior derivative that, when acting on a kform produces a (k+1)form. This operation extends the differential of a function, and the divergence and the curl of a vector field in an appropriate sense that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the general Stokes' theorem. In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as De Rham's theorem.
The general setting for the study of differential forms is on a differentiable manifold. Differential 1forms are naturally dual to vector fields on a manifold, and the pairing between vector fields and 1forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided the information is expressed in terms of differential forms. As an example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.
History
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan in his 1899 paper.^{[1]}
Concept
Differential forms provide an approach to multivariable calculus that is independent of coordinates.
Let U be an open set in R^{n}. A differential 0form ("zero form") is defined to be a smooth function f on U. If v is any vector in R^{n}, then f has a directional derivative ∂_{v} f, which is another function on U whose value at a point p ∈ U is the rate of change (at p) of f in the v direction:
 $(\backslash partial\_v\; f)(p)\; =\; \backslash frac\{d\}\{dt\}\; f(p+tv)\backslash Big\_\{t=0\}.$
(This notion can be extended to the case that v is a vector field on U by evaluating v at the point p in the definition.)
In particular, if v = e_{j} is the jth coordinate vector then ∂_{v}f is the partial derivative of f with respect to the jth coordinate function, i.e., ∂f / ∂x^{j}, where x^{1}, x^{2}, ... x^{n} are the coordinate functions on U. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates y^{1}, y^{2}, ... y^{n} are introduced, then
 $\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x^j\}\; =\; \backslash sum\_\{i=1\}^n\backslash frac\{\backslash partial\; y^i\}\{\backslash partial\; x^j\}\backslash frac\{\backslash partial\; f\}\{\backslash partial\; y^i\}$
The first idea leading to differential forms is the observation that ∂_{v} f (p) is a linear function of v:
 $(\backslash partial\_\{v+w\}\; f)(p)\; =\; (\backslash partial\_v\; f)(p)\; +\; (\backslash partial\_w\; f)(p)$
 $(\backslash partial\_\{c\; v\}\; f)(p)\; =\; c\; (\backslash partial\_v\; f)(p)$
for any vectors v, w and any real number c. This linear map from R^{n} to R is denoted df_{p} and called the derivative of f at p. Thus df_{p}(v) = ∂_{v} f (p). The object df can be viewed as a function on U, whose value at p is not a real number, but the linear map df_{p}. This is just the usual Fréchet derivative — an example of a differential 1form.
Since any vector v is a linear combination ∑ v^{j}e_{j} of its components, df is uniquely determined by df_{p}(e_{j}) for each j and each p∈U, which are just the partial derivatives of f on U. Thus df provides a way of encoding the partial derivatives of f. It can be decoded by noticing that the coordinates x^{1}, x^{2},... x^{n} are themselves functions on U, and so define differential 1forms dx^{1}, dx^{2}, ..., dx^{n}. Since ∂x^{i} / ∂x^{j} = δ_{ij}, the Kronecker delta function, it follows that

The meaning of this expression is given by evaluating both sides at an arbitrary point p: on the right hand side, the sum is defined "pointwise", so that
 $df\_p\; =\; \backslash sum\_\{i=1\}^n\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x^i\}(p)\; (dx^i)\_p.$
Applying both sides to e_{j}, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula (*).
More generally, for any smooth functions g_{i} and h_{i} on U, we define the differential 1form α = ∑_{i} g_{i} dh_{i} pointwise by
 $\backslash alpha\_p\; =\; \backslash sum\_i\; g\_i(p)\; (dh\_i)\_p\backslash ,\backslash !$
for each p ∈ U. Any differential 1form arises this way, and by using (*) it follows that any differential 1form α on U may be expressed in coordinates as
 $\backslash alpha\; =\; \backslash sum\_\{i=1\}^n\; f\_i\backslash ,\; dx^i$
for some smooth functions f_{i} on U.
The second idea leading to differential forms arises from the following question: given a differential 1form α on U, when does there exist a function f on U such that α = df? The above expansion reduces this question to the search for a function f whose partial derivatives ∂f / ∂x^{i} are equal to n given functions f_{i}. For n>1, such a function does not always exist: any smooth function f satisfies
 $\backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; x^i\; \backslash ,\; \backslash partial\; x^j\}\; =\; \backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; x^j\; \backslash ,\; \backslash partial\; x^i\}$
so it will be impossible to find such an f unless
 $\backslash frac\{\backslash partial\; f\_j\}\{\backslash partial\; x^i\}\; \; \backslash frac\{\backslash partial\; f\_i\}\{\backslash partial\; x^j\}=0.$
for all i and j.
The skewsymmetry of the left hand side in i and j suggests introducing an antisymmetric product $\backslash wedge$ on differential 1forms, the wedge product, so that these equations can be combined into a single condition
 $\backslash sum\_\{i,j=1\}^n\; \backslash frac\{\backslash partial\; f\_j\}\{\backslash partial\; x^i\}\; dx^i\; \backslash wedge\; dx^j\; =\; 0$
where
 $dx^i\; \backslash wedge\; dx^j\; =\; \; dx^j\; \backslash wedge\; dx^i.\; \backslash ,$
This is an example of a differential 2form: the exterior derivative dα of α= ∑_{j=1}^{n} f_{j} dx^{j} is given by
 $d\backslash alpha\; =\; \backslash sum\_\{j=1\}^n\; df\_j\; \backslash wedge\; dx^j\; =\; \backslash sum\_\{i,j=1\}^n\; \backslash frac\{\backslash partial\; f\_j\}\{\backslash partial\; x^i\}\; dx^i\; \backslash wedge\; dx^j.$
To summarize: dα = 0 is a necessary condition for the existence of a function f with α = df.
Differential 0forms, 1forms, and 2forms are special cases of differential forms. For each k, there is a space of differential kforms, which can be expressed in terms of the coordinates as
 $\backslash sum\_\{i\_1,i\_2\backslash ldots\; i\_k=1\}^n\; f\_\{i\_1i\_2\backslash ldots\; i\_k\}\; dx^\{i\_1\}\; \backslash wedge\; dx^\{i\_2\}\; \backslash wedge\backslash cdots\; \backslash wedge\; dx^\{i\_k\}$
for a collection of functions f_{i1i2 ... ik}. (Of course, as assumed below, one can restrict the sum to the case $i\_1\backslash cdots\{k1\}\backslash ,\; math>.)$
Differential forms can be multiplied together using the wedge product, and for any differential kform α, there is a differential (k + 1)form dα called the exterior derivative of α.
Differential forms, the wedge product and the exterior derivative are independent of a choice of coordinates. Consequently they may be defined on any smooth manifold M. One way to do this is cover M with coordinate charts and define a differential kform on M to be a family of differential kforms on each chart which agree on the overlaps. However, there are more intrinsic definitions which make the independence of coordinates manifest.
Intrinsic definitions
Let M be a smooth manifold. A differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of M. At any point p∈M, a kform β defines an alternating multilinear map
 $\backslash beta\_p\backslash colon\; T\_p\; M\backslash times\; \backslash cdots\; \backslash times\; T\_p\; M\; \backslash to\; \backslash mathbb\{R\}$
(with k factors of T_{p}M in the product), where T_{p}M is the tangent space to M at p. Equivalently, β is a totally antisymmetric covariant tensor field of rank k.
The set of all differential kforms on a manifold M is a vector space, often denoted Ω^{k}(M).
For example, a differential 1form α assigns to each point p∈M a linear functional α_{p} on T_{p}M. In the presence of an inner product on T_{p}M (induced by a Riemannian metric on M), α_{p} may be represented as the inner product with a tangent vector X_{p}. Differential 1forms are sometimes called covariant vector fields, covector fields, or "dual vector fields", particularly within physics.
Operations
There are several operations on differential forms: the wedge product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, and the Lie derivative of a differential form with respect to a vector field.
Wedge product
The wedge product of a kform α and an lform β is a (k + l)form denoted αΛβ. For example, if k = l = 1, then αΛβ is the 2form whose value at a point p is the alternating bilinear form defined by
 $(\backslash alpha\backslash wedge\backslash beta)\_p(v,w)=\backslash alpha\_p(v)\backslash beta\_p(w)\; \; \backslash alpha\_p(w)\backslash beta\_p(v)$
for v, w ∈ T_{p}M. (In an alternative convention, the right hand side is divided by two in this formula.)
The wedge product is bilinear: for instance, if α, β, and γ are any differential forms, then
 $\backslash alpha\; \backslash wedge\; (\backslash beta\; +\; \backslash gamma)\; =\; \backslash alpha\; \backslash wedge\; \backslash beta\; +\; \backslash alpha\; \backslash wedge\; \backslash gamma.$
It is skew commutative (also known as graded commutative), meaning that it satisfies a variant of anticommutativity that depends on the degrees of the forms: if α is a kform and β is an lform, then
 $\backslash alpha\; \backslash wedge\; \backslash beta\; =\; (1)^\{kl\}\; \backslash beta\; \backslash wedge\; \backslash alpha.\; \backslash ,$
Riemannian manifold
On a Riemannian manifold, or more generally a pseudoRiemannian manifold, vector fields and covector field can be identified (the metric is a fiberwise isomorphism of the tangent space and the cotangent space), and additional operations can thus be defined, such as the Hodge star operator $*\backslash colon\; \backslash Omega^k(M)\; \backslash overset\{\backslash sim\}\{\backslash to\}\; \backslash Omega^\{nk\}(M)$ and codifferential $\backslash delta\backslash colon\; \backslash Omega^k(M)\backslash rightarrow\; \backslash Omega^\{k1\}(M),$ (degree $1$) which is adjoint to the exterior differential d.
Vector field structures
On a pseudoRiemannian manifold, 1forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion.
Firstly, each (co)tangent space generates a Clifford algebra, where the product of a (co)vector with itself is given by the value of a quadratic form  in this case, the natural one induced by the metric. This algebra is distinct from the exterior algebra of differential forms, which can be viewed as a Clifford algebra where the quadratic form vanishes (since the exterior product of any vector with itself is zero). Clifford algebras are thus nonanticommutative ("quantum") deformations of the exterior algebra. They are studied in geometric algebra.
Another alternative is to consider vector fields as derivations, and consider the (noncommutative) algebra of differential operators they generate, which is the Weyl algebra, and is a noncommutative ("quantum") deformation of the symmetric algebra in the vector fields.
Exterior differential complex
One important property of the exterior derivative is that d^{2} = 0. This means that the exterior derivative defines a cochain complex:
 $0\; \backslash to\backslash Omega^0(M)\backslash \; \backslash stackrel\{d\}\{\backslash to\}\backslash \; \backslash Omega^1(M)\backslash \; \backslash stackrel\{d\}\{\backslash to\}\backslash \; \backslash Omega^2(M)\backslash \; \backslash stackrel\{d\}\{\backslash to\}\backslash \; \backslash Omega^3(M)\; \backslash to\; \backslash cdots\; \backslash \; \backslash to\backslash \; \backslash Omega^n(M)\backslash \; \backslash to\; \backslash \; 0.$
By the Poincaré lemma, this complex is locally exact except at Ω^{0}(M). Its cohomology is the de Rham cohomology of M.
Pullback
One of the main reasons the cotangent bundle rather than the tangent bundle is used in the construction of the exterior complex is that differential forms are capable of being pulled back by smooth maps, while vector fields cannot be pushed forward by smooth maps unless the map is, say, a diffeomorphism. The existence of pullback homomorphisms in de Rham cohomology depends on the pullback of differential forms.
Differential forms can be moved from one manifold to another using a smooth map. If f : M → N is smooth and ω is a smooth kform on N, then there is a differential form f^{*}ω on M, called the pullback of ω, which captures the behavior of ω as seen relative to f.
To define the pullback, recall that the differential of f is a map f_{*} : TM → TN. Fix a differential kform ω on N. For a point p of M and tangent vectors v_{1}, ..., v_{k} to M at p, the pullback of ω is defined by the formula
 $(f^*\backslash omega)\_p(v\_1,\; \backslash ldots,\; v\_k)\; =\; \backslash omega\_\{f(p)\}(f\_*v\_1,\; \backslash ldots,\; f\_*v\_k).$
More abstractly, if ω is viewed as a section of the cotangent bundle T^{*}N of N, then f^{*}ω is the section of T^{*}M defined as the composite map
 $M\; \backslash stackrel\{f\}\{\backslash to\}\; N\; \backslash stackrel\{\backslash omega\}\{\backslash to\}\; T^*N\; \backslash stackrel\{(Df)^*\}\{\backslash longrightarrow\}\; T^*M.$
Pullback respects all of the basic operations on forms:
 $f^*(\backslash omega\; +\; \backslash eta)\; =\; f^*\backslash omega\; +\; f^*\backslash eta,$
 $f^*(\backslash omega\backslash wedge\backslash eta)\; =\; f^*\backslash omega\backslash wedge\; f^*\backslash eta,$
 $f^*(d\backslash omega)\; =\; d(f^*\backslash omega).$
The pullback of a form can also be written in coordinates. Assume that x_{1}, ..., x_{m} are coordinates on M, that y_{1}, ..., y_{n} are coordinates on N, and that these coordinate systems are related by the formulas y_{i} = f_{i}(x_{1}, ..., x_{m}) for all i. Then, locally on N, ω can be written as
 $\backslash omega\; =\; \backslash sum\_\{i\_1\; <\; \backslash cdots\; <\; i\_k\}\; \backslash omega\_\{i\_1\backslash cdots\; i\_k\}dy\_\{i\_1\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dy\_\{i\_k\},$
where, for each choice of i_{1}, ..., i_{k}, $\backslash omega\_\{i\_1\backslash cdots\; i\_k\}$ is a realvalued function of y_{1}, ..., y_{n}. Using the linearity of pullback and its compatibility with wedge product, the pullback of ω has the formula
 $f^*\backslash omega\; =\; \backslash sum\_\{i\_1\; <\; \backslash cdots\; <\; i\_k\}\; (\backslash omega\_\{i\_1\backslash cdots\; i\_k\}\backslash circ\; f)df\_\{i\_1\}\backslash wedge\backslash cdots\backslash wedge\; df\_\{i\_n\}.$
Each exterior derivative df_{i} can be expanded in terms of dx_{1}, ..., dx_{m}. The resulting kform can be written using Jacobian matrices:
 $f^*\backslash omega\; =\; \backslash sum\_\{i\_1\; <\; \backslash cdots\; <\; i\_k\}\; \backslash sum\_\{j\_1\; <\; \backslash cdots\; <\; j\_k\}\; (\backslash omega\_\{i\_1\backslash cdots\; i\_k\}\backslash circ\; f)\backslash frac\{\backslash partial(f\_\{i\_1\},\; \backslash ldots,\; f\_\{i\_k\})\}\{\backslash partial(x\_\{j\_1\},\; \backslash ldots,\; x\_\{j\_k\})\}dx\_\{j\_1\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx\_\{j\_k\}.$
Here, $\backslash frac\{\backslash partial(f\_\{i\_1\},\; \backslash ldots,\; f\_\{i\_k\})\}\{\backslash partial(x\_\{j\_1\},\; \backslash ldots,\; x\_\{j\_k\})\}$ stands for the determinant of the matrix whose entries are $\backslash partial\; f\_\{i\_m\}/\backslash partial\; x\_\{j\_n\}$, $1\backslash leq\; m,n\backslash leq\; k$.
Integration
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. Simply, a chain parametrizes a domain of integration as a collection of cells (images of cubes or other domains D) that are patched together; to integrate, one pulls back the form on each cell of the chain to a form on the cube (or other domain) and integrates there, which is just integration of a function on $\backslash mathbf\{R\}^k,$ as the pulled back form is simply a multiple of the volume form $du^1\; \backslash cdots\; du^k.$ For example, given a path $\backslash gamma(t)\; \backslash colon\; [0,1]\; \backslash to\; \backslash mathbf\{R\}^2,$ integrating a form on the path is simply pulling back the form to a function on $[0,1]$ (properly, to a form $f(t)\backslash ,dt$) and integrating the function on the interval.
Let
 $\backslash omega=\backslash sum\_\{i\_1\; <\; \backslash cdots\; <\; i\_k\}\; a\_\{i\_1,\backslash dots,i\_k\}(\{\backslash mathbf\; x\})\backslash ,dx^\{i\_1\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^\{i\_k\}$
be a differential form and S a differentiable kmanifold over which we wish to integrate, where S has the parameterization
 $S(\{\backslash mathbf\; u\})=(x^1(\{\backslash mathbf\; u\}),\backslash dots,x^n(\{\backslash mathbf\; u\}))$
for u in the parameter domain D. Then (Rudin 1976) defines the integral of the differential form over S as
 $\backslash int\_S\; \backslash omega\; =\backslash int\_D\; \backslash sum\_\{i\_1\; <\; \backslash cdots\; <\; i\_k\}\; a\_\{i\_1,\backslash dots,i\_k\}(S(\{\backslash mathbf\; u\}))\; \backslash frac\{\backslash partial(x^\{i\_1\},\backslash dots,x^\{i\_k\})\}\{\backslash partial(u^\{1\},\backslash dots,u^\{k\})\}\backslash ,du^1\backslash ldots\; du^k$
where
 $\backslash frac\{\backslash partial(x^\{i\_1\},\backslash dots,x^\{i\_k\})\}\{\backslash partial(u^\{1\},\backslash dots,u^\{k\})\}$
is the determinant of the Jacobian. The Jacobian exists because S is differentiable.
More generally, a $k$form can be integrated over an $p$dimensional submanifold, for $p\backslash leq\; k$, to obtain a $(kp)$form. This comes up, for example, in defining the pushforward of a differential form by a smooth map $f:\; M\backslash to\; N$ by attempting to integrate over the fibers of $f$.
Stokes' theorem
The fundamental relationship between the exterior derivative and integration is given by the general Stokes theorem: If $\backslash omega$ is an n−1form with compact support on M and ∂M denotes the boundary of M with its induced orientation, then
 $\backslash int\_M\; d\backslash omega\; =\; \backslash oint\_\{\backslash partial\; M\}\; \backslash omega.\backslash !\backslash ,$
A key consequence of this is that "the integral of a closed form over homologous chains is equal": if $\backslash omega$ is a closed kform and M and N are kchains that are homologous (such that MN is the boundary of a (k+1)chain W), then $\backslash textstyle\{\backslash int\_M\; \backslash omega\; =\; \backslash int\_N\; \backslash omega\},$ since the difference is the integral $\backslash textstyle\{\backslash int\_W\; d\backslash omega\; =\; \backslash int\_W\; 0\; =\; 0\}.$
For example, if $\backslash omega\; =\; df$ is the derivative of a potential function on the plane or $\backslash mathbf\{R\}^n,$ then the integral of $\backslash omega$ over a path from a to b does not depend on the choice of path (the integral is $f(b)f(a)$), since different paths with given endpoints are homotopic, hence homologous (a weaker condition). This case is called the gradient theorem, and generalizes the fundamental theorem of calculus). This path independence is very useful in contour integration.
This theorem also underlies the duality between de Rham cohomology and the homology of chains.
Relation with measures
On a general differentiable manifold (without additional structure), differential forms cannot be integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains, and measures, which are integrated over subsets. The simplest example is attempting to integrate the 1form dx over the interval [0,1]. Assuming the usual distance (and thus measure) on the real line, this integral is either 1 or −1, depending on orientation: $\backslash textstyle\{\backslash int\_0^1\; dx\; =\; 1\},$ while $\backslash textstyle\{\backslash int\_1^0\; dx\; =\; \; \backslash int\_0^1\; dx\; =\; 1\}.$ By contrast, the integral of the measure dx on the interval is unambiguously 1 (formally, the integral of the constant function 1 with respect to this measure is 1). Similarly, under a change of coordinates a differential nform changes by the Jacobian determinant J, while a measure changes by the absolute value of the Jacobian determinant, $J,$ which further reflects the issue of orientation. For example, under the map $x\; \backslash mapsto\; x$ on the line, the differential form $dx$ pulls back to $dx;$ orientation has reversed; while the Lebesgue measure, also denoted $dx,$ pulls back to $dx;$ it does not change.
In the presence of the additional data of an orientation, it is possible to integrate nforms (topdimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over the fundamental class of the manifold, $[M].$ Formally, in the presence of an orientation, one may identify nforms with densities on a manifold; densities in turn define a measure, and thus can be integrated (Folland 1999, Section 11.4, pp. 361–362).
On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate nforms over compact subsets, with the two choices differing by a sign. On nonorientable manifold, nforms and densities cannot be identified —notably, any topdimensional form must vanish somewhere (there are no volume forms on nonorientable manifolds), but there are nowherevanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate nforms. One can instead identify densities with topdimensional pseudoforms.
There is in general no meaningful way to integrate kforms over subsets for $k\; <\; n$ because there is no consistent way to orient kdimensional subsets; geometrically, a kdimensional subset can be turned around in place, reversing any orientation but yielding the same subset. Compare the Gram determinant of a set of k vectors in an ndimensional space, which, unlike the determinant of n vectors, is always positive, corresponding to a squared number.
On a Riemannian manifold, one may define a kdimensional Hausdorff measure for any k (integer or real), which may be integrated over kdimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over kdimensional subsets, providing a measuretheoretic analog to integration of kforms. The ndimensional Hausdorff measure yields a density, as above.
Applications in physics
Differential forms arise in some important physical contexts. For example, in Maxwell's theory of electromagnetism, the Faraday 2form, or electromagnetic field strength, is
 $\backslash textbf\{F\}\; =\; \backslash frac\{1\}\{2\}f\_\{ab\}\backslash ,\; dx^a\; \backslash wedge\; dx^b\backslash ,,$
where the $f\_\{ab\}$ are formed from the electromagnetic fields $\backslash vec\; E$ and $\backslash vec\; B$, e.g. $f\_\{1,2\}=E\_z/c\backslash ,,$ $\backslash ,f\_\{2,3\}=B\_z$, or equivalent definitions.
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 connection form for the principal bundle is the vector potential, typically denoted by A, when represented in some gauge. One then has
 $\backslash textbf\{F\}\; =\; d\backslash textbf\{A\}.$
The current 3form is
 $\backslash textbf\{J\}\; =\; \backslash frac\{1\}\{6\}\; j^a\backslash ,\; \backslash epsilon\_\{abcd\}\backslash ,\; dx^b\; \backslash wedge\; dx^c\; \backslash wedge\; dx^d\backslash ,,$
where $j^a$ are the four components of the currentdensity. (Here it is a matter of convention, to write $\backslash ,F\_\{ab\}$ instead of $\backslash ,f\_\{ab\}\backslash ,,$ i.e. to use capital letters, and to write $J^a$ instead of $j^a$. However, the vector rsp. tensor components and the abovementioned forms have different physical dimensions. Moreover, one should remember that by decision of an international commission of the IUPAP, the magnetic polarization vector is called $\backslash vec\; J$ since several decades, and by some publishers $\backslash mathbf\; J\backslash ,,$ i.e. the same name is used for totally different quantities.)
Using the abovementioned definitions, Maxwell's equations can be written very compactly in geometrized units as
 $d\backslash ,\; \{\backslash textbf\{F\}\}\; =\; \backslash textbf\{0\}$
 $d\backslash ,\; \{*\backslash textbf\{F\}\}\; =\; \backslash textbf\{J\}$
where $*$ denotes the Hodge star operator. Similar considerations describe the geometry of gauge theories in general.
The 2form $*\; \backslash mathbf\{F\}\backslash ,,$ which is dual to the Faraday form, is also called Maxwell 2form.
Electromagnetism is an example of a U(1) gauge theory. Here the Lie group is U(1), the onedimensional unitary group, which is in particular abelian. There are gauge theories, such as YangMills theory, in which the Lie group is not abelian. In that case, one gets relations which are similar to those described here. The analog of the field F in such theories is the curvature form of the connection, which is represented in a gauge by a Lie algebravalued oneform A. The YangMills field F is then defined by
 $\backslash mathbf\{F\}\; =\; d\backslash mathbf\{A\}\; +\; \backslash mathbf\{A\}\backslash wedge\backslash mathbf\{A\}.$
In the abelian case, such as electromagnetism, $\backslash mathbf\; A\backslash wedge\; \backslash mathbf\; A=0$, but this does not hold in general. Likewise the field equations are modified by additional terms involving wedge products of A and F, owing to the structure equations of the gauge group.
Applications in geometric measure theory
Numerous minimality results for complex analytic manifolds are based on the Wirtinger inequality for 2forms. 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
References


 —Translation of Formes différentielles (1967)








External links
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