Differential form

Inmathematics,differential formsprovide a unified approach to defineintegrandsover curves, surfaces, solids, and higher-dimensionalmanifolds.The modern notion of differential forms was pioneered byÉlie Cartan.It has many applications, especially in geometry, topology and physics.

For instance, the expression $f (x) dx$ is an example of a $1$ -form,and can beintegratedover an interval $[a, b]$ contained in the domain of $f$ :

\int _{a}^{b}f(x)\,dx.

Similarly, the expression $f (x, y, z) dx \land dy + g (x, y, z) dz \land dx + h (x, y, z) dy \land dz$ is a $2$ -formthat can be integrated over asurface $S$ :

\int _{S}(f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge dx+h(x,y,z)\,dy\wedge dz).

The symbol $\land$ denotes theexterior product,sometimes called thewedge product,of two differential forms. Likewise, a $3$ -form $f (x, y, z) dx \land dy \land dz$ represents avolume elementthat can be integrated over a region of space. In general, a $k$ -form is an object that may be integrated over a $k$ -dimensional manifold, and ishomogeneousof degree $k$ in the coordinate differentials $dx,dy,\ldots.$ On an $n$ -dimensional manifold, the top-dimensional form ( $n$ -form) is called avolume form.

The differential forms form analternating algebra.This implies that $dy\wedge dx=-dx\wedge dy$ and $dx\wedge dx=0.$ This alternating property reflects theorientationof the domain of integration.

Theexterior derivativeis an operation on differential forms that, given a $k$ -form $\varphi$ ,produces a $(k +1)$ -form $d\varphi.$ This operation extends thedifferential of a function(a function can be considered as a $0$ -form, and its differential is $df(x)=f'(x)dx$ ). This allows expressing thefundamental theorem of calculus,thedivergence theorem,Green's theorem,andStokes' theoremas special cases of a single general result, thegeneralized Stokes theorem.

Differential $1$ -forms are naturally dual tovector fieldson adifferentiable manifold,and the pairing between vector fields and $1$ -forms is extended to arbitrary differential forms by theinterior product.The algebra of differential forms along with the exterior derivative defined on it is preserved by thepullbackunder smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, thechange of variables formulafor 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 toÉlie Cartanwith reference to his 1899 paper.^[1]Some aspects of theexterior algebraof differential forms appears inHermann Grassmann's 1844 work,Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).

Concept

Differential forms provide an approach tomultivariable calculusthat is independent ofcoordinates.

Integration and orientation

A differential $k$ -form can be integrated over an orientedmanifoldof dimension $k$ .A differential $1$ -form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential $2$ -form can be thought of as measuring an infinitesimal oriented area, or 2-dimensional oriented density. And so on.

Integration of differential forms is well-defined only onoriented manifolds.An example of a 1-dimensional manifold is an interval $[a, b]$ ,and intervals can be given an orientation: they are positively oriented if $a < b$ ,and negatively oriented otherwise. If $a < b$ then the integral of the differential $1$ -form $f (x) dx$ over the interval $[a, b]$ (with its natural positive orientation) is

\int _{a}^{b}f(x)\,dx

which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is:

\int _{b}^{a}f(x)\,dx=-\int _{a}^{b}f(x)\,dx.

This gives a geometrical context to theconventionsfor one-dimensional integrals, that the sign changes when the orientation of the interval is reversed. A standard explanation of this in one-variable integration theory is that, when the limits of integration are in the opposite order ( $b < a$ ), the increment $dx$ is negative in the direction of integration.

More generally, an $m$ -form is an oriented density that can be integrated over an $m$ -dimensional oriented manifold. (For example, a $1$ -form can be integrated over an oriented curve, a $2$ -form can be integrated over an oriented surface, etc.) If $M$ is an oriented $m$ -dimensional manifold, and $M'$ is the same manifold with opposite orientation and $ω$ is an $m$ -form, then one has:

\int _{M}\omega =-\int _{M'}\omega \,.

These conventions correspond to interpreting the integrand as a differential form, integrated over achain.Inmeasure theory,by contrast, one interprets the integrand as a function $f$ with respect to a measure $μ$ and integrates over a subset $A$ ,without any notion of orientation; one writes ${\textstyle \int _{A}f\,d\mu =\int _{[a,b]}f\,d\mu }$ to indicate integration over a subset $A$ .This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; seebelowfor details.

Making the notion of an oriented density precise, and thus of a differential form, involves theexterior algebra.The differentials of a set of coordinates, $dx 1$ ,..., $dx n$ can be used as a basis for all $1$ -forms. Each of these represents acovectorat each point on the manifold that may be thought of as measuring a small displacement in the corresponding coordinate direction. A general $1$ -form is a linear combination of these differentials at every point on the manifold:

f_{1}\,dx^{1}+\cdots +f_{n}\,dx^{n},

where the $f k = f k (x 1,..., x n)$ are functions of all the coordinates. A differential $1$ -form is integrated along an oriented curve as a line integral.

The expressions $dx i \land dx j$ ,where $i < j$ can be used as a basis at every point on the manifold for all $2$ -forms. This may be thought of as an infinitesimal oriented square parallel to the $x i$ – $x j$ -plane. A general $2$ -form is a linear combination of these at every point on the manifold: ${\textstyle \sum _{1\leq i<j\leq n}f_{i,j}\,dx^{i}\wedge dx^{j}}$ ,and it is integrated just like a surface integral.

A fundamental operation defined on differential forms is theexterior product(the symbol is thewedge $\land$ ). This is similar to thecross productfrom vector calculus, in that it is an alternating product. For instance,

dx^{1}\wedge dx^{2}=-dx^{2}\wedge dx^{1}

because the square whose first side is $dx 1$ and second side is $dx 2$ is to be regarded as having the opposite orientation as the square whose first side is $dx 2$ and whose second side is $dx 1$ .This is why we only need to sum over expressions $dx i \land dx j$ ,with $i < j$ ;for example: $a (dx i \land dx j) + b (dx j \land dx i) = (a - b) dx i \land dx j$ .The exterior product allows higher-degree differential forms to be built out of lower-degree ones, in much the same way that thecross productin vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also implies that $dx i \land dx i = 0$ ,in the same way that the cross product of parallel vectors, whose magnitude is the area of the parallelogram spanned by those vectors, is zero. In higher dimensions, $dx i 1 \land \cdot\cdot\cdot \land dx i m = 0$ if any two of the indices $i 1$ ,..., $i m$ are equal, in the same way that the "volume" enclosed by aparallelotopewhose edge vectors arelinearly dependentis zero.

Multi-index notation

A common notation for the wedge product of elementary $k$ -forms is so calledmulti-index notation:in an $n$ -dimensional context, for $I=(i_{1},i_{2},\ldots,i_{k}),1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n$ ,we define ${\textstyle dx^{I}:=dx^{i_{1}}\wedge \cdots \wedge dx^{i_{k}}=\bigwedge _{i\in I}dx^{i}}$ .^[2]Another useful notation is obtained by defining the set of all strictly increasing multi-indices of length $k$ ,in a space of dimension $n$ ,denoted ${\mathcal {J}}_{k,n}:=\{I=(i_{1},\ldots,i_{k}):1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n\}$ .Then locally (wherever the coordinates apply), $\{dx^{I}\}_{I\in {\mathcal {J}}_{k,n}}$ spans the space of differential $k$ -forms in a manifold $M$ of dimension $n$ ,when viewed as a module over the ring $C \infty (M)$ of smooth functions on $M$ .By calculating the size of ${\mathcal {J}}_{k,n}$ combinatorially, the module of $k$ -forms on an $n$ -dimensional manifold, and in general space of $k$ -covectors on an $n$ -dimensional vector space, is $n$ choose $k$ : ${\textstyle |{\mathcal {J}}_{k,n}|={\binom {n}{k}}}$ .This also demonstrates that there are no nonzero differential forms of degree greater than the dimension of the underlying manifold.

The exterior derivative

In addition to the exterior product, there is also theexterior derivativeoperator $d$ .The exterior derivative of a differential form is a generalization of thedifferential of a function,in the sense that the exterior derivative of $f \in C \infty (M) = Ω 0 (M)$ is exactly the differential of $f$ .When generalized to higher forms, if $ω = f dx I$ is a simple $k$ -form, then its exterior derivative $dω$ is a $(k + 1)$ -form defined by taking the differential of the coefficient functions:

d\omega =\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}\wedge dx^{I}.

with extension to general $k$ -forms through linearity: if ${\textstyle \tau =\sum _{I\in {\mathcal {J}}_{k,n}}a_{I}\,dx^{I}\in \Omega ^{k}(M)}$ ,then its exterior derivative is

d\tau =\sum _{I\in {\mathcal {J}}_{k,n}}\left(\sum _{j=1}^{n}{\frac {\partial a_{I}}{\partial x^{j}}}\,dx^{j}\right)\wedge dx^{I}\in \Omega ^{k+1}(M)

In $R 3$ ,with theHodge star operator,the exterior derivative corresponds togradient,curl,anddivergence,although this correspondence, like the cross product, does not generalize to higher dimensions, and should be treated with some caution.

The exterior derivative itself applies in an arbitrary finite number of dimensions, and is a flexible and powerful tool with wide application indifferential geometry,differential topology,and many areas in physics. Of note, although the above definition of the exterior derivative was defined with respect to local coordinates, it can be defined in an entirely coordinate-free manner, as anantiderivationof degree 1 on theexterior algebraof differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate onmanifolds.It also allows for a natural generalization of thefundamental theorem of calculus,called the (generalized)Stokes' theorem,which is a central result in the theory of integration on manifolds.

Differential calculus

Let $U$ be anopen setin $R n$ .A differential $0$ -form ( "zero-form" ) is defined to be asmooth function $f$ on $U$ – the set of which is denoted $C \infty (U)$ .If $v$ is any vector in $R n$ ,then $f$ has adirectional derivative $\partial v f$ ,which is another function on $U$ whose value at a point $p \in U$ is the rate of change (at $p$ ) of $f$ in the $v$ direction:

(\partial _{\mathbf {v} }f)(p)=\left.{\frac {d}{dt}}f(p+t\mathbf {v} )\right|_{t=0}.

(This notion can be extended pointwise to the case that $v$ is avector fieldon $U$ by evaluating $v$ at the point $p$ in the definition.)

In particular, if $v = e j$ is the $j$ thcoordinate vectorthen $\partial v f$ is thepartial derivativeof $f$ with respect to the $j$ th coordinate vector, i.e., $\partial f / \partial x j$ ,where $x 1$ , $x 2$ ,..., $x n$ are the coordinate vectors in $U$ .By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates $y 1$ , $y 2$ ,..., $y n$ are introduced, then

{\frac {\partial f}{\partial x^{j}}}=\sum _{i=1}^{n}{\frac {\partial y^{i}}{\partial x^{j}}}{\frac {\partial f}{\partial y^{i}}}.

The first idea leading to differential forms is the observation that $\partial v f (p)$ is alinear functionof $v$ :

{\begin{aligned}(\partial _{\mathbf {v} +\mathbf {w} }f)(p)&=(\partial _{\mathbf {v} }f)(p)+(\partial _{\mathbf {w} }f)(p)\\(\partial _{c\mathbf {v} }f)(p)&=c(\partial _{\mathbf {v} }f)(p)\end{aligned}}

for any vectors $v$ , $w$ and any real number $c$ .At each pointp,thislinear mapfrom $R n$ to $R$ is denoted $df p$ and called thederivativeordifferentialof $f$ at $p$ .Thus $df p (v) = \partial v f (p)$ .Extended over the whole set, the object $df$ can be viewed as a function that takes a vector field on $U$ ,and returns a real-valued function whose value at each point is the derivative along the vector field of the function $f$ .Note that at each $p$ ,the differential $df p$ is not a real number, but a linear functional on tangent vectors, and a prototypical example of a differential $1$ -form.

Since any vector $v$ is alinear combination $Σ v j e j$ of itscomponents, $df$ is uniquely determined by $df p (e j)$ for each $j$ and each $p \in U$ ,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 $1$ -forms $dx 1$ , $dx 2$ ,..., $dx n$ .Let $f = x i$ .Since $\partial x i / \partial x j = δ ij$ ,theKronecker delta function,it follows that

df=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}\,dx^{i}.

(*)

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}=\sum _{i=1}^{n}{\frac {\partial f}{\partial x^{i}}}(p)(dx^{i})_{p}.

Applying both sides to $e j$ ,the result on each side is the $j$ th 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 $1$ -form $α = Σ i g i dh i$ pointwise by

\alpha _{p}=\sum _{i}g_{i}(p)(dh_{i})_{p}

for each $p \in U$ .Any differential $1$ -form arises this way, and by using(*)it follows that any differential $1$ -form $α$ on $U$ may be expressed in coordinates as

\alpha =\sum _{i=1}^{n}f_{i}\,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 $1$ -form $α$ 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 $\partial f / \partial x i$ are equal to $n$ given functions $f i$ .For $n > 1$ ,such a function does not always exist: any smooth function $f$ satisfies

{\frac {\partial ^{2}f}{\partial x^{i}\,\partial x^{j}}}={\frac {\partial ^{2}f}{\partial x^{j}\,\partial x^{i}}},

so it will be impossible to find such an $f$ unless

{\frac {\partial f_{j}}{\partial x^{i}}}-{\frac {\partial f_{i}}{\partial x^{j}}}=0

for all $i$ and $j$ .

Theskew-symmetryof the left hand side in $i$ and $j$ suggests introducing an antisymmetric product $\land$ on differential $1$ -forms, theexterior product,so that these equations can be combined into a single condition

\sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}=0,

where $\land$ is defined so that:

dx^{i}\wedge dx^{j}=-dx^{j}\wedge dx^{i}.

This is an example of a differential $2$ -form. This $2$ -form is called theexterior derivative $dα$ of $α = Σ n j =1 f j dx j$ .It is given by

d\alpha =\sum _{j=1}^{n}df_{j}\wedge dx^{j}=\sum _{i,j=1}^{n}{\frac {\partial f_{j}}{\partial x^{i}}}\,dx^{i}\wedge dx^{j}.

To summarize: $dα = 0$ is a necessary condition for the existence of a function $f$ with $α = df$ .

Differential $0$ -forms, $1$ -forms, and $2$ -forms are special cases of differential forms. For each $k$ ,there is a space of differential $k$ -forms, which can be expressed in terms of the coordinates as

\sum _{i_{1},i_{2}\ldots i_{k}=1}^{n}f_{i_{1}i_{2}\ldots i_{k}}\,dx^{i_{1}}\wedge dx^{i_{2}}\wedge \cdots \wedge dx^{i_{k}}

for a collection of functions $f i 1 i 2 \cdot\cdot\cdot i k$ .Antisymmetry, which was already present for $2$ -forms, makes it possible to restrict the sum to those sets of indices for which $i 1 < i 2 <... < i k -1 < i k$ .

Differential forms can be multiplied together using the exterior product, and for any differential $k$ -form $α$ ,there is a differential $(k + 1)$ -form $dα$ called the exterior derivative of $α$ .

Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates. Consequently, they may be defined on anysmooth manifold $M$ .One way to do this is cover $M$ withcoordinate chartsand define a differential $k$ -form on $M$ to be a family of differential $k$ -forms 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 asmooth manifold.A smooth differential form of degree $k$ is asmooth sectionof the $k$ thexterior powerof thecotangent bundleof $M$ .The set of all differential $k$ -forms on a manifold $M$ is avector space,often denoted $Ω k (M)$ .

The definition of a differential form may be restated as follows. At any point $p \in M$ ,a $k$ -form $β$ defines an element

\beta _{p}\in {\textstyle \bigwedge }^{k}T_{p}^{*}M,

where $T p M$ is thetangent spaceto $M$ at $p$ and $T p * M$ is itsdual space.This space isnaturally isomorphic^[3]^{[clarification needed]}to the fiber at $p$ of the dual bundle of the $k$ th exterior power of thetangent bundleof $M$ .That is, $β$ is also a linear functional ${\textstyle \beta _{p}\colon {\textstyle \bigwedge }^{k}T_{p}M\to \mathbf {R} }$ ,i.e. the dual of the $k$ th exterior power is isomorphic to the $k$ th exterior power of the dual:

{\textstyle \bigwedge }^{k}T_{p}^{*}M\cong {\Big (}{\textstyle \bigwedge }^{k}T_{p}M{\Big )}^{*}

By the universal property of exterior powers, this is equivalently analternating multilinear map:

\beta _{p}\colon \bigoplus _{n=1}^{k}T_{p}M\to \mathbf {R}.

Consequently, a differential $k$ -form may be evaluated against any $k$ -tuple of tangent vectors to the same point $p$ of $M$ .For example, a differential $1$ -form $α$ assigns to each point $p \in M$ alinear functional $α p$ on $T p M$ .In the presence of aninner producton $T p M$ (induced by aRiemannian metricon $M$ ), $α p$ may berepresentedas the inner product with atangent vector $X p$ .Differential $1$ -forms are sometimes calledcovariant vector fields,covector fields, or "dual vector fields", particularly within physics.

The exterior algebra may be embedded in the tensor algebra by means of the alternation map. The alternation map is defined as a mapping

\operatorname {Alt} \colon {\bigotimes }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.

For a tensor $\tau$ at a point $p$ ,

\operatorname {Alt} (\tau _{p})(x_{1},\dots,x_{k})={\frac {1}{k!}}\sum _{\sigma \in S_{k}}\operatorname {sgn}(\sigma )\tau _{p}(x_{\sigma (1)},\dots,x_{\sigma (k)}),

where $S k$ is thesymmetric groupon $k$ elements. The alternation map is constant on the cosets of the ideal in the tensor algebra generated by the symmetric 2-forms, and therefore descends to an embedding

\operatorname {Alt} \colon {\textstyle \bigwedge }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.

This map exhibits $β$ as atotally antisymmetric covariant tensor fieldof rank $k$ .The differential forms on $M$ are in one-to-one correspondence with such tensor fields.

Operations

As well as the addition and multiplication by scalar operations which arise from the vector space structure, there are several other standard operations defined on differential forms. The most important operations are theexterior productof two differential forms, theexterior derivativeof a single differential form, theinterior productof a differential form and a vector field, theLie derivativeof a differential form with respect to a vector field and thecovariant derivativeof a differential form with respect to a vector field on a manifold with a defined connection.

Exterior product

The exterior product of a $k$ -form $α$ and an $ℓ$ -form $β$ ,denoted $α \land β$ ,is a ( $k + ℓ$ )-form. At each point $p$ of the manifold $M$ ,the forms $α$ and $β$ are elements of an exterior power of the cotangent space at $p$ .When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra).

The antisymmetry inherent in the exterior algebra means that when $α \land β$ is viewed as a multilinear functional, it is alternating. However, when the exterior algebra is embedded as a subspace of the tensor algebra by means of the alternation map, the tensor product $α \otimes β$ is not alternating. There is an explicit formula which describes the exterior product in this situation. The exterior product is

\alpha \wedge \beta =\operatorname {Alt} (\alpha \otimes \beta ).

If the embedding of ${\textstyle \bigwedge }^{n}T^{*}M$ into ${\bigotimes }^{n}T^{*}M$ is done via the map $n!\operatorname {Alt}$ instead of $\operatorname {Alt}$ ,the exterior product is

\alpha \wedge \beta ={\frac {(k+\ell )!}{k!\ell!}}\operatorname {Alt} (\alpha \otimes \beta ).

This description is useful for explicit computations. For example, if $k = ℓ = 1$ ,then $α \land β$ is the $2$ -form whose value at a point $p$ is thealternating bilinear formdefined by

(\alpha \wedge \beta )_{p}(v,w)=\alpha _{p}(v)\beta _{p}(w)-\alpha _{p}(w)\beta _{p}(v)

for $v, w \in T p M$ .

The exterior product is bilinear: If $α$ , $β$ ,and $γ$ are any differential forms, and if $f$ is any smooth function, then

\alpha \wedge (\beta +\gamma )=\alpha \wedge \beta +\alpha \wedge \gamma,

\alpha \wedge (f\cdot \beta )=f\cdot (\alpha \wedge \beta ).

It isskew commutative(also known asgraded commutative), meaning that it satisfies a variant ofanticommutativitythat depends on the degrees of the forms: if $α$ is a $k$ -form and $β$ is an $ℓ$ -form, then

\alpha \wedge \beta =(-1)^{k\ell }\beta \wedge \alpha.

One also has thegraded Leibniz rule:

$d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1)^{k}\alpha \wedge d\beta.$

Riemannian manifold

On aRiemannian manifold,or more generally apseudo-Riemannian manifold,the metric defines a fibre-wise isomorphism of the tangent and cotangent bundles. This makes it possible to convert vector fields to covector fields and vice versa. It also enables the definition of additional operations such as theHodge star operator $\star \colon \Omega ^{k}(M)\ {\stackrel {\sim }{\to }}\ \Omega ^{n-k}(M)$ and thecodifferential $\delta \colon \Omega ^{k}(M)\rightarrow \Omega ^{k-1}(M)$ ,which has degree $-1$ and isadjointto the exterior differential $d$ .

Vector field structures

On a pseudo-Riemannian manifold, $1$ -forms 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 aClifford 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 themetric.This algebra isdistinctfrom theexterior algebraof 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 non-anticommutative ( "quantum" ) deformations of the exterior algebra. They are studied ingeometric algebra.

Another alternative is to consider vector fields as derivations. The (noncommutative) algebra ofdifferential operatorsthey generate is theWeyl algebraand is a noncommutative ( "quantum" ) deformation of thesymmetricalgebra 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 acochain complex:

0\ \to \ \Omega ^{0}(M)\ {\stackrel {d}{\to }}\ \Omega ^{1}(M)\ {\stackrel {d}{\to }}\ \Omega ^{2}(M)\ {\stackrel {d}{\to }}\ \Omega ^{3}(M)\ \to \ \cdots \ \to \ \Omega ^{n}(M)\ \to \ 0.

This complex is called the de Rham complex, and itscohomologyis by definition thede Rham cohomologyof $M$ .By thePoincaré lemma,the de Rham complex is locallyexactexcept at $Ω 0 (M)$ .The kernel at $Ω 0 (M)$ is the space oflocally constant functionson $M$ .Therefore, the complex is a resolution of the constantsheaf $R$ ,which in turn implies a form of de Rham's theorem: de Rham cohomology computes thesheaf cohomologyof $R$ .

Pullback

Suppose that $f : M \to N$ is smooth. The differential of $f$ is a smooth map $df : TM \to TN$ between the tangent bundles of $M$ and $N$ .This map is also denoted $f *$ and called thepushforward.For any point $p \in M$ and any tangent vector $v \in T p M$ ,there is a well-defined pushforward vector $f * (v)$ in $T f (p) N$ .However, the same is not true of a vector field. If $f$ is not injective, say because $q \in N$ has two or more preimages, then the vector field may determine two or more distinct vectors in $T q N$ .If $f$ is not surjective, then there will be a point $q \in N$ at which $f *$ does not determine any tangent vector at all. Since a vector field on $N$ determines, by definition, a unique tangent vector at every point of $N$ ,the pushforward of a vector field does not always exist.

By contrast, it is always possible to pull back a differential form. A differential form on $N$ may be viewed as a linear functional on each tangent space. Precomposing this functional with the differential $df : TM \to TN$ defines a linear functional on each tangent space of $M$ and therefore a differential form on $M$ .The existence of pullbacks is one of the key features of the theory of differential forms. It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology.

Formally, let $f : M \to N$ be smooth, and let $ω$ be a smooth $k$ -form on $N$ .Then there is a differential form $f * ω$ on $M$ ,called thepullbackof $ω$ ,which captures the behavior of $ω$ as seen relative to $f$ .To define the pullback, fix 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^{*}\omega )_{p}(v_{1},\ldots,v_{k})=\omega _{f(p)}(f_{*}v_{1},\ldots,f_{*}v_{k}).

There are several more abstract ways to view this definition. If $ω$ is a $1$ -form on $N$ ,then it may be viewed as a section of the cotangent bundle $T * N$ of $N$ .Using^∗to denote a dual map, the dual to the differential of $f$ is $(df) * : T * N \to T * M$ .The pullback of $ω$ may be defined to be the composite

M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ T^{*}N\ {\stackrel {(df)^{*}}{\longrightarrow }}\ T^{*}M.

This is a section of the cotangent bundle of $M$ and hence a differential $1$ -form on $M$ .In full generality, let ${\textstyle \bigwedge ^{k}(df)^{*}}$ denote the $k$ th exterior power of the dual map to the differential. Then the pullback of a $k$ -form $ω$ is the composite

M\ {\stackrel {f}{\to }}\ N\ {\stackrel {\omega }{\to }}\ {\textstyle \bigwedge }^{k}T^{*}N\ {\stackrel {{\bigwedge }^{k}(df)^{*}}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}T^{*}M.

Another abstract way to view the pullback comes from viewing a $k$ -form $ω$ as a linear functional on tangent spaces. From this point of view, $ω$ is a morphism ofvector bundles

{\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R},

where $N \times R$ is the trivial rank one bundle on $N$ .The composite map

{\textstyle \bigwedge }^{k}TM\ {\stackrel {{\bigwedge }^{k}df}{\longrightarrow }}\ {\textstyle \bigwedge }^{k}TN\ {\stackrel {\omega }{\to }}\ N\times \mathbf {R}

defines a linear functional on each tangent space of $M$ ,and therefore it factors through the trivial bundle $M \times R$ .The vector bundle morphism ${\textstyle {\textstyle \bigwedge }^{k}TM\to M\times \mathbf {R} }$ defined in this way is $f * ω$ .

Pullback respects all of the basic operations on forms. If $ω$ and $η$ are forms and $c$ is a real number, then

{\begin{aligned}f^{*}(c\omega )&=c(f^{*}\omega ),\\f^{*}(\omega +\eta )&=f^{*}\omega +f^{*}\eta,\\f^{*}(\omega \wedge \eta )&=f^{*}\omega \wedge f^{*}\eta,\\f^{*}(d\omega )&=d(f^{*}\omega ).\end{aligned}}

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 $i$ .Locally on $N$ , $ω$ can be written as

\omega =\sum _{i_{1}<\cdots <i_{k}}\omega _{i_{1}\cdots i_{k}}\,dy^{i_{1}}\wedge \cdots \wedge dy^{i_{k}},

where, for each choice of $i 1$ ,..., $i k$ , $ω i 1 \cdot\cdot\cdot i k$ is a real-valued function of $y 1$ ,..., $y n$ .Using the linearity of pullback and its compatibility with exterior product, the pullback of $ω$ has the formula

f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f)\,df_{i_{1}}\wedge \cdots \wedge df_{i_{k}}.

Each exterior derivative $df i$ can be expanded in terms of $dx 1$ ,..., $dx m$ .The resulting $k$ -form can be written usingJacobianmatrices:

f^{*}\omega =\sum _{i_{1}<\cdots <i_{k}}\sum _{j_{1}<\cdots <j_{k}}(\omega _{i_{1}\cdots i_{k}}\circ f){\frac {\partial (f_{i_{1}},\ldots,f_{i_{k}})}{\partial (x^{j_{1}},\ldots,x^{j_{k}})}}\,dx^{j_{1}}\wedge \cdots \wedge dx^{j_{k}}.

Here, ${\textstyle {\frac {\partial (f_{i_{1}},\ldots,f_{i_{k}})}{\partial (x^{j_{1}},\ldots,x^{j_{k}})}}}$ denotes the determinant of the matrix whose entries are ${\textstyle {\frac {\partial f_{i_{m}}}{\partial x^{j_{n}}}}}$ , $1\leq m,n\leq k$ .

Integration

A differential $k$ -form can be integrated over an oriented $k$ -dimensional manifold. When the $k$ -form is defined on an $n$ -dimensional manifold with $n > k$ ,then the $k$ -form can be integrated over oriented $k$ -dimensional submanifolds. If $k = 0$ ,integration over oriented 0-dimensional submanifolds is just the summation of the integrand evaluated at points, according to the orientation of those points. Other values of $k = 1, 2, 3,...$ correspond to line integrals, surface integrals, volume integrals, and so on. There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of Euclidean space.

Integration on Euclidean space

Let $U$ be an open subset of $R n$ .Give $R n$ its standard orientation and $U$ the restriction of that orientation. Every smooth $n$ -form $ω$ on $U$ has the form

\omega =f(x)\,dx^{1}\wedge \cdots \wedge dx^{n}

for some smooth function $f : R n \to R$ .Such a function has an integral in the usual Riemann or Lebesgue sense. This allows us to define the integral of $ω$ to be the integral of $f$ :

\int _{U}\omega \ {\stackrel {\text{def}}{=}}\int _{U}f(x)\,dx^{1}\cdots dx^{n}.

Fixing an orientation is necessary for this to be well-defined. The skew-symmetry of differential forms means that the integral of, say, $dx 1 \land dx 2$ must be the negative of the integral of $dx 2 \land dx 1$ .Riemann and Lebesgue integrals cannot see this dependence on the ordering of the coordinates, so they leave the sign of the integral undetermined. The orientation resolves this ambiguity.

Integration over chains

Let $M$ be an $n$ -manifold and $ω$ an $n$ -form on $M$ .First, assume that there is a parametrization of $M$ by an open subset of Euclidean space. That is, assume that there exists a diffeomorphism

\varphi \colon D\to M

where $D \subseteq R n$ .Give $M$ the orientation induced by $φ$ .Then (Rudin 1976) defines the integral of $ω$ over $M$ to be the integral of $φ * ω$ over $D$ .In coordinates, this has the following expression. Fix an embedding of $M$ in $R I$ with coordinates $x 1,..., x I$ .Then

\omega =\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots,i_{n}}({\mathbf {x} })\,dx^{i_{1}}\wedge \cdots \wedge dx^{i_{n}}.

Suppose that $φ$ is defined by

\varphi ({\mathbf {u} })=(x^{1}({\mathbf {u} }),\ldots,x^{I}({\mathbf {u} })).

Then the integral may be written in coordinates as

\int _{M}\omega =\int _{D}\sum _{i_{1}<\cdots <i_{n}}a_{i_{1},\ldots,i_{n}}(\varphi ({\mathbf {u} })){\frac {\partial (x^{i_{1}},\ldots,x^{i_{n}})}{\partial (u^{1},\dots,u^{n})}}\,du^{1}\cdots du^{n},

where

{\frac {\partial (x^{i_{1}},\ldots,x^{i_{n}})}{\partial (u^{1},\ldots,u^{n})}}

is the determinant of theJacobian.The Jacobian exists because $φ$ is differentiable.

In general, an $n$ -manifold cannot be parametrized by an open subset of $R n$ .But such a parametrization is always possible locally, so it is possible to define integrals over arbitrary manifolds by defining them as sums of integrals over collections of local parametrizations. Moreover, it is also possible to define parametrizations of $k$ -dimensional subsets for $k < n$ ,and this makes it possible to define integrals of $k$ -forms. To make this precise, it is convenient to fix a standard domain $D$ in $R k$ ,usually a cube or a simplex. A $k$ -chainis a formal sum of smooth embeddings $D \to M$ .That is, it is a collection of smooth embeddings, each of which is assigned an integer multiplicity. Each smooth embedding determines a $k$ -dimensional submanifold of $M$ .If the chain is

c=\sum _{i=1}^{r}m_{i}\varphi _{i},

then the integral of a $k$ -form $ω$ over $c$ is defined to be the sum of the integrals over the terms of $c$ :

\int _{c}\omega =\sum _{i=1}^{r}m_{i}\int _{D}\varphi _{i}^{*}\omega.

This approach to defining integration does not assign a direct meaning to integration over the whole manifold $M$ .However, it is still possible to assign such a meaning indirectly because every smooth manifold may be smoothlytriangulatedin an essentially unique way, and the integral over $M$ may be defined to be the integral over the chain determined by a triangulation.

Integration using partitions of unity

There is another approach, expounded in (Dieudonné 1972), which does directly assign a meaning to integration over $M$ ,but this approach requires fixing an orientation of $M$ .The integral of an $n$ -form $ω$ on an $n$ -dimensional manifold is defined by working in charts. Suppose first that $ω$ is supported on a single positively oriented chart. On this chart, it may be pulled back to an $n$ -form on an open subset of $R n$ .Here, the form has a well-defined Riemann or Lebesgue integral as before. The change of variables formula and the assumption that the chart is positively oriented together ensure that the integral of $ω$ is independent of the chosen chart. In the general case, use a partition of unity to write $ω$ as a sum of $n$ -forms, each of which is supported in a single positively oriented chart, and define the integral of $ω$ to be the sum of the integrals of each term in the partition of unity.

It is also possible to integrate $k$ -forms on oriented $k$ -dimensional submanifolds using this more intrinsic approach. The form is pulled back to the submanifold, where the integral is defined using charts as before. For example, given a path $γ (t): [0, 1] \to R 2$ ,integrating a $1$ -form on the path is simply pulling back the form to a form $f (t) dt$ on $[0, 1]$ ,and this integral is the integral of the function $f (t)$ on the interval.

Integration along fibers

Fubini's theoremstates that the integral over a set that is a product may be computed as an iterated integral over the two factors in the product. This suggests that the integral of a differential form over a product ought to be computable as an iterated integral as well. The geometric flexibility of differential forms ensures that this is possible not just for products, but in more general situations as well. Under some hypotheses, it is possible to integrate along the fibers of a smooth map, and the analog of Fubini's theorem is the case where this map is the projection from a product to one of its factors.

Because integrating a differential form over a submanifold requires fixing an orientation, a prerequisite to integration along fibers is the existence of a well-defined orientation on those fibers. Let $M$ and $N$ be two orientable manifolds of pure dimensions $m$ and $n$ ,respectively. Suppose that $f : M \to N$ is a surjective submersion. This implies that each fiber $f -1 (y)$ is $(m - n)$ -dimensional and that, around each point of $M$ ,there is a chart on which $f$ looks like the projection from a product onto one of its factors. Fix $x \in M$ and set $y = f (x)$ .Suppose that

{\begin{aligned}\omega _{x}&\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,\\\eta _{y}&\in {\textstyle \bigwedge }^{n}T_{y}^{*}N,\end{aligned}}

and that $η y$ does not vanish. Following (Dieudonné 1972), there is a unique

\sigma _{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}(f^{-1}(y))

which may be thought of as the fibral part of $ω x$ with respect to $η y$ .More precisely, define $j : f -1 (y) \to M$ to be the inclusion. Then $σ x$ is defined by the property that

\omega _{x}=(f^{*}\eta _{y})_{x}\wedge \sigma '_{x}\in {\textstyle \bigwedge }^{m}T_{x}^{*}M,

where

\sigma '_{x}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M

is any $(m - n)$ -covector for which

\sigma _{x}=j^{*}\sigma '_{x}.

The form $σ x$ may also be notated $ω x / η y$ .

Moreover, for fixed $y$ , $σ x$ varies smoothly with respect to $x$ .That is, suppose that

\omega \colon f^{-1}(y)\to T^{*}M

is a smooth section of the projection map; we say that $ω$ is a smooth differential $m$ -form on $M$ along $f -1 (y)$ .Then there is a smooth differential $(m - n)$ -form $σ$ on $f -1 (y)$ such that, at each $x \in f -1 (y)$ ,

\sigma _{x}=\omega _{x}/\eta _{y}.

This form is denoted $ω / η y$ .The same construction works if $ω$ is an $m$ -form in a neighborhood of the fiber, and the same notation is used. A consequence is that each fiber $f -1 (y)$ is orientable. In particular, a choice of orientation forms on $M$ and $N$ defines an orientation of every fiber of $f$ .

The analog of Fubini's theorem is as follows. As before, $M$ and $N$ are two orientable manifolds of pure dimensions $m$ and $n$ ,and $f : M \to N$ is a surjective submersion. Fix orientations of $M$ and $N$ ,and give each fiber of $f$ the induced orientation. Let $ω$ be an $m$ -form on $M$ ,and let $η$ be an $n$ -form on $N$ that is almost everywhere positive with respect to the orientation of $N$ .Then, for almost every $y \in N$ ,the form $ω / η y$ is a well-defined integrable $m - n$ form on $f -1 (y)$ .Moreover, there is an integrable $n$ -form on $N$ defined by

y\mapsto {\bigg (}\int _{f^{-1}(y)}\omega /\eta _{y}{\bigg )}\,\eta _{y}.

Denote this form by

{\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta.

Then (Dieudonné 1972) proves the generalized Fubini formula

\int _{M}\omega =\int _{N}{\bigg (}\int _{f^{-1}(y)}\omega /\eta {\bigg )}\,\eta.

It is also possible to integrate forms of other degrees along the fibers of a submersion. Assume the same hypotheses as before, and let $α$ be a compactly supported $(m - n + k)$ -form on $M$ .Then there is a $k$ -form $γ$ on $N$ which is the result of integrating $α$ along the fibers of $f$ .The form $α$ is defined by specifying, at each $y \in N$ ,how $γ$ pairs with each $k$ -vector $v$ at $y$ ,and the value of that pairing is an integral over $f -1 (y)$ that depends only on $α$ , $v$ ,and the orientations of $M$ and $N$ .More precisely, at each $y \in N$ ,there is an isomorphism

{\textstyle \bigwedge }^{k}T_{y}N\to {\textstyle \bigwedge }^{n-k}T_{y}^{*}N

defined by the interior product

\mathbf {v} \mapsto \mathbf {v} \,\lrcorner \,\zeta _{y},

for any choice of volume form $ζ$ in the orientation of $N$ .If $x \in f -1 (y)$ ,then a $k$ -vector $v$ at $y$ determines an $(n - k)$ -covector at $x$ by pullback:

f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\in {\textstyle \bigwedge }^{n-k}T_{x}^{*}M.

Each of these covectors has an exterior product against $α$ ,so there is an $(m - n)$ -form $β v$ on $M$ along $f -1 (y)$ defined by

(\beta _{\mathbf {v} })_{x}=\left(\alpha _{x}\wedge f^{*}(\mathbf {v} \,\lrcorner \,\zeta _{y})\right){\big /}\zeta _{y}\in {\textstyle \bigwedge }^{m-n}T_{x}^{*}M.

This form depends on the orientation of $N$ but not the choice of $ζ$ .Then the $k$ -form $γ$ is uniquely defined by the property

\langle \gamma _{y},\mathbf {v} \rangle =\int _{f^{-1}(y)}\beta _{\mathbf {v} },

and $γ$ is smooth (Dieudonné 1972). This form also denoted $α ♭$ and called theintegral of $α$ along the fibers of $f$ .Integration along fibers is important for the construction of Gysin maps in de Rham cohomology.

Integration along fibers satisfies theprojection formula(Dieudonné 1972). If $λ$ is any $ℓ$ -form on $N$ ,then

\alpha ^{\flat }\wedge \lambda =(\alpha \wedge f^{*}\lambda )^{\flat }.

Stokes's theorem

The fundamental relationship between the exterior derivative and integration is given by theStokes' theorem:If $ω$ is an ( $n - 1$ )-form with compact support on $M$ and $\partialM$ denotes theboundaryof $M$ with its inducedorientation,then

\int _{M}d\omega =\int _{\partial M}\omega.

A key consequence of this is that "the integral of a closed form over homologous chains is equal": If $ω$ is a closed $k$ -form and $M$ and $N$ are $k$ -chains that are homologous (such that $M - N$ is the boundary of a $(k + 1)$ -chain $W$ ), then $\textstyle {\int _{M}\omega =\int _{N}\omega }$ ,since the difference is the integral $\textstyle \int _{W}d\omega =\int _{W}0=0$ .

For example, if $ω = df$ is the derivative of a potential function on the plane or $R n$ ,then the integral of $ω$ 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 arehomotopic,hence homologous (a weaker condition). This case is called thegradient theorem,and generalizes thefundamental theorem of calculus.This path independence is very useful incontour integration.

This theorem also underlies the duality betweende Rham cohomologyand thehomologyof chains.

Relation with measures

On ageneraldifferentiable manifold (without additional structure), differential formscannotbe integrated over subsets of the manifold; this distinction is key to the distinction between differential forms, which are integrated over chains or oriented submanifolds, and measures, which are integrated over subsets. The simplest example is attempting to integrate the $1$ -form $dx$ over the interval $[0, 1]$ .Assuming the usual distance (and thus measure) on the real line, this integral is either $1$ or $-1$ ,depending onorientation: $\textstyle {\int _{0}^{1}dx=1}$ ,while $\textstyle {\int _{1}^{0}dx=-\int _{0}^{1}dx=-1}$ .By contrast, the integral of themeasure $| dx |$ on the interval is unambiguously $1$ (i.e. the integral of the constant function $1$ with respect to this measure is $1$ ). Similarly, under a change of coordinates a differential $n$ -form changes by theJacobian determinant $J$ ,while a measure changes by theabsolute valueof the Jacobian determinant, $| J |$ ,which further reflects the issue of orientation. For example, under the map $x \mapsto - x$ on the line, the differential form $dx$ pulls back to $- dx$ ;orientation has reversed; while theLebesgue measure,which here we denote $| dx |$ ,pulls back to $| dx |$ ;it does not change.

In the presence of the additional data of anorientation,it is possible to integrate $n$ -forms (top-dimensional forms) over the entire manifold or over compact subsets; integration over the entire manifold corresponds to integrating the form over thefundamental classof the manifold, $[M]$ .Formally, in the presence of an orientation, one may identify $n$ -forms withdensities 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 $n$ -forms over compact subsets, with the two choices differing by a sign. On non-orientable manifold, $n$ -forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are novolume formson non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate $n$ -forms. One can instead identify densities with top-dimensionalpseudoforms.

Even in the presence of an orientation, there is in general no meaningful way to integrate $k$ -forms over subsets for $k < n$ because there is no consistent way to use the ambient orientation to orient $k$ -dimensional subsets. Geometrically, a $k$ -dimensional subset can be turned around in place, yielding the same subset with the opposite orientation; for example, the horizontal axis in a plane can be rotated by 180 degrees. Compare theGram determinantof a set of $k$ vectors in an $n$ -dimensional space, which, unlike the determinant of $n$ vectors, is always positive, corresponding to a squared number. An orientation of a $k$ -submanifold is therefore extra data not derivable from the ambient manifold.

On a Riemannian manifold, one may define a $k$ -dimensionalHausdorff measurefor any $k$ (integer or real), which may be integrated over $k$ -dimensional subsets of the manifold. A function times this Hausdorff measure can then be integrated over $k$ -dimensional subsets, providing a measure-theoretic analog to integration of $k$ -forms. The $n$ -dimensional Hausdorff measure yields a density, as above.

Currents

The differential form analog of adistributionor generalized function is called acurrent.The space of $k$ -currents on $M$ is the dual space to an appropriate space of differential $k$ -forms. Currents play the role of generalized domains of integration, similar to but even more flexible than chains.

Applications in physics

Differential forms arise in some important physical contexts. For example, in Maxwell's theory ofelectromagnetism,theFaraday 2-form,orelectromagnetic field strength,is

{\textbf {F}}={\frac {1}{2}}f_{ab}\,dx^{a}\wedge dx^{b}\,,

where the $f ab$ are formed from the electromagnetic fields ${\vec {E}}$ and ${\vec {B}}$ ;e.g., $f 12 = E z / c$ , $f 23 = - B z$ ,or equivalent definitions.

This form is a special case of thecurvature formon the $U(1)$ principal bundleon which both electromagnetism and generalgauge theoriesmay be described. Theconnection formfor the principal bundle is the vector potential, typically denoted by $A$ ,when represented in some gauge. One then has

{\textbf {F}}=d{\textbf {A}}.

Thecurrent $3$ -formis

{\textbf {J}}={\frac {1}{6}}j^{a}\,\varepsilon _{abcd}\,dx^{b}\wedge dx^{c}\wedge dx^{d}\,,

where $j a$ are the four components of the current density. (Here it is a matter of convention to write $F ab$ instead of $f ab$ ,i.e. to use capital letters, and to write $J a$ instead of $j a$ .However, the vector rsp. tensor components and the above-mentioned forms have different physical dimensions. Moreover, by decision of an international commission of theInternational Union of Pure and Applied Physics,the magnetic polarization vector has been called ${\vec {J}}$ for several decades, and by some publishers $J$ ;i.e., the same name is used for different quantities.)

Using the above-mentioned definitions,Maxwell's equationscan be written very compactly ingeometrized unitsas

{\begin{aligned}d{\textbf {F}}&={\textbf {0}}\\d{\star {\textbf {F}}}&={\textbf {J}},\end{aligned}}

where $\star$ denotes theHodge staroperator. Similar considerations describe the geometry of gauge theories in general.

The $2$ -form ${\star }\mathbf {F}$ ,which isdualto the Faraday form, is also calledMaxwell 2-form.

Electromagnetism is an example of a $U(1)$ gauge theory.Here theLie groupis $U(1)$ ,the one-dimensionalunitary group,which is in particularabelian.There are gauge theories, such asYang–Mills 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 aLie algebra-valued one-form $A$ .The Yang–Mills field $F$ is then defined by

\mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A}.

In the abelian case, such as electromagnetism, $A \land A = 0$ ,but this does not hold in general. Likewise the field equations are modified by additional terms involving exterior products of $A$ and $F$ ,owing to thestructure equationsof the gauge group.

Applications in geometric measure theory

Numerous minimality results for complex analytic manifolds are based on theWirtinger inequality for 2-forms.A succinct proof may be found inHerbert Federer's classic textGeometric Measure Theory.The Wirtinger inequality is also a key ingredient inGromov's inequality for complex projective spaceinsystolic geometry.

Notes

^Cartan, Élie (1899),"Sur certaines expressions différentielles et le problème de Pfaff",Annales Scientifiques de l'École Normale Supérieure,16:239–332,doi:10.24033/asens.467
^Tu, Loring W. (2011).An introduction to manifolds(2nd ed.). New York: Springer.ISBN 9781441974006.OCLC 682907530.
^"Linear algebra –" Natural "pairings between exterior powers of a vector space and its dual".

References

Bachman, David (2006),A Geometric Approach to Differential Forms,Birkhäuser,ISBN 978-0-8176-4499-4
Bachman, David (2003),A Geometric Approach to Differential Forms,arXiv:math/0306194v1,Bibcode:2003math......6194B
Cartan, Henri(2006),Differential Forms,Dover,ISBN 0-486-45010-4—Translation ofFormes différentielles(1967)
Dieudonné, Jean(1972),Treatise on Analysis,vol. 3, New York-London: Academic Press, Inc.,MR 0350769
Edwards, Harold M.(1994),Advanced Calculus; A Differential Forms Approach,Modern Birkhäuser Classics, Boston, Basel, Berlin: Birkhäuser,doi:10.1007/978-0-8176-8412-9,ISBN 978-0-8176-8411-2
Folland, Gerald B.(1999),Real Analysis: Modern Techniques and Their Applications(Second ed.),ISBN 978-0-471-31716-6,provides a brief discussion of integration on manifolds from the point of view of measure theory in the last section.
Flanders, Harley(1989) [1964],Differential forms with applications to the physical sciences,Mineola, New York: Dover Publications,ISBN 0-486-66169-5
Fleming, Wendell H.(1965), "Chapter 6: Exterior algebra and differential calculus",Functions of Several Variables,Addison-Wesley, pp. 205–238.This textbook inmultivariate calculusintroduces the exterior algebra of differential forms at the college calculus level.
Morita, Shigeyuki (2001),Geometry of Differential Forms,AMS,ISBN 0-8218-1045-6
Rudin, Walter(1976),Principles of Mathematical Analysis,New York: McGraw-Hill,ISBN 0-07-054235-X
Spivak, Michael(1965),Calculus on Manifolds,Menlo Park, California: W. A. Benjamin,ISBN 0-8053-9021-9,standard introductory text.
Tu, Loring W.(2008),An Introduction to Manifolds,Universitext, Springer,doi:10.1007/978-1-4419-7400-6,ISBN 978-0-387-48098-5
Zorich, Vladimir A.(2004),Mathematical Analysis II,Springer,ISBN 3-540-40633-6

External links

Weisstein, Eric W."Differential form".MathWorld.
Sjamaar, Reyer (2006),Manifolds and differential forms lecture notes(PDF),a course taught atCornell University.
Bachman, David (2003),A Geometric Approach to Differential Forms,arXiv:math/0306194,Bibcode:2003math......6194B,an undergraduate text.
Needham, Tristan.Visual differential geometry and forms: a mathematical drama in five acts.Princeton University Press, 2021.

[1] Cartan, Élie (1899),"Sur certaines expressions différentielles et le problème de Pfaff",Annales Scientifiques de l'École Normale Supérieure,16:239–332,doi:10.24033/asens.467

[2] Tu, Loring W. (2011).An introduction to manifolds(2nd ed.). New York: Springer.ISBN 9781441974006.OCLC 682907530.

[3] "Linear algebra –" Natural "pairings between exterior powers of a vector space and its dual".

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[3]