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Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus which is independent of coordinates. A differential form of degree k, or (differential) k-form, on a smooth manifold M is a smooth section of the kth exterior power of the cotangent bundle of M. The set of all k-forms on M is a vector space commonly denoted Ωk(M).

A differential 0-form is by definition a smooth function on M. A differential 1-form is an object dual to a vector field on M.

Differential forms can be multiplied together using an operation called the wedge product. There is also a differential operator on differential forms called the exterior derivative. The wedge product of a k-form and an l-form is a (k+l)-form, and the exterior derivative of a k-form is a (k+1)-form. In particular, the exterior derivative of a 0-form (which is a function on M) is its differential (which is a 1-form on M).

The modern notion of differential forms was pioneered by Élie Cartan, and has many applications, especially in geometry, topology and physics.

Concept

Differential forms provide an approach to multivariable calculus which is independent of coordinates.

Let U be an open set in Rn. A differential 0-form ("zero form") is defined to be a smooth function f on U. If v is any vector in Rn, then f has a directional derivativev f, which is another function on U whose value at a point pU is the rate of change (at p) of f in the v direction:

$\left(partial_v f\right)\left(p\right) = frac\left\{mathrm d\right\}\left\{ mathrm d t\right\} f\left(p+tv\right)Big|_\left\{t=0\right\}.$
(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=ej is the jth coordinate vector then ∂vf is the partial derivative of f with respect to the jth coordinate function, i.e., ∂f / ∂xj, where x1, x2,... xn are the coordinate functions on U. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates y1 , y2,... yn are introduced, then

$frac\left\{partial f\right\}\left\{partial x^j\right\} = sum_\left\{i=1\right\}^nfrac\left\{partial y^i\right\}\left\{partial x^j\right\}frac\left\{partial f\right\}\left\{partial y^i\right\}$

The first idea leading to differential forms is the observation that ∂v f (p) is a linear function of v:

$\left(partial_\left\{v+w\right\} f\right)\left(p\right) = \left(partial_v f\right)\left(p\right) + \left(partial_w f\right)\left(p\right)$
$\left(partial_\left\{c v\right\} f\right)\left(p\right) = c \left(partial_v f\right)\left(p\right)$
for any vectors v, w and any real number c. This linear map from Rn to R is denoted dfp and called the derivative of f at p. Thus dfp(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 dfp. This is an example of a differential 1-form.

Since any vector v is a linear combinationvjej of its components, df is uniquely determined by dfp(ej) for each j and each pU, 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 x1, x2,... xn are themselves functions on U, and so define differential 1-forms dx1, dx2,... dxn. Since ∂xi / ∂xj = δij, the Kronecker delta function, it follows that

$\left(*\right)quad mathrm d f = sum_\left\{i=1\right\}^n frac\left\{partial f\right\}\left\{partial x^i\right\} mathrm\left\{d\right\} x^i.$

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

$mathrm d f_p = sum_\left\{i=1\right\}^n frac\left\{partial f\right\}\left\{partial x^i\right\}\left(p\right) \left(mathrm\left\{d\right\} x^i\right)_p.$
Applying both sides to ej, 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 gi and hi on U, we define the differential 1-form α = ∑i gi dhi pointwise by

$alpha_p = sum_i g_i\left(p\right) \left(mathrm d h_i\right)_p,!$
for each pU. 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
$alpha = sum_\left\{i=1\right\}^n f_i, mathrm\left\{d\right\} x^i$
for some smooth functions fi 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 ∂f / ∂xi are equal to n given functions fi. For n>1, such a function does not always exist: any smooth function f satisfies

$frac\left\{partial^2 f\right\}\left\{partial x^ipartial x^j\right\} = frac\left\{partial^2 f\right\}\left\{partial x^jpartial x^i\right\}$
so it will be impossible to find such an f unless
$frac\left\{partial f_j\right\}\left\{partial x^i\right\} - frac\left\{partial f_i\right\}\left\{partial x^j\right\}=0.$
for all i and j.

The skew-symmetry of the left hand side in i and j suggests introducing an antisymmetric product $wedge$ on differential 1-forms, the wedge product, so that these equations can be combined into a single condition

$sum_\left\{i,j=1\right\}^n frac\left\{partial f_j\right\}\left\{partial x^i\right\} \left\{mathrm d\right\}x^i wedge \left\{mathrm d\right\}x^j = 0$
where
$\left\{mathrm d\right\}x^i wedge \left\{mathrm d\right\}x^j = - \left\{mathrm d\right\}x^j wedge \left\{mathrm d\right\}x^i.$
This is an example of a differential 2-form: the exterior derivative dα of α= ∑j=1n fj dxj is given by
$mathrm dalpha = sum_\left\{j=1\right\}^n mathrm d f_j wedge dx^j = sum_\left\{i,j=1\right\}^n frac\left\{partial f_j\right\}\left\{partial x^i\right\} \left\{mathrm d\right\}x^i wedge \left\{mathrm d\right\}x^j.$
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

$sum_\left\{i_1,i_2ldots i_k=1\right\}^n f_\left\{i_1i_2ldots i_k\right\} mathrm d x^\left\{i_1\right\} wedgemathrm d x^\left\{i_1\right\} wedgecdots wedge mathrm d x^\left\{i_k\right\}$
for a collection of functions fi1i2...ik.

Differential forms can be multiplied together using the wedge 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 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 k-form on M to be a 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 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 pM, a k-form β defines an alternating multilinear map

$beta_pcolon T_p Mtimes cdots times T_p M to mathbb\left\{R\right\}$
(with k factors of TpM in the product), where TpM 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 k-forms on a manifold M is a vector space, often denoted Ωk(M).

For example, a differential 1-form α assigns to each point pM a linear functional αp on TpM. In the presence of an inner product on TpM (induced by a Riemannian metric on M), αp may be represented as the inner product with a tangent vector Xp. Differential 1-forms are sometimes called covariant vector fields, covector fields, or "dual vector fields", particular within physics.

Operations

There are several operations on differential forms: wedge product, exterior derivative, interior product and Lie derivative.

The wedge product of a k-form α and an l-form β is a (k+l)-form denoted αΛβ. For example, if k=l=1, then αΛβ is the 2-form whose value at a point p is the alternating bilinear form defined by

$\left(alphawedgebeta\right)_p\left(v,w\right)=alpha_p\left(v\right)beta_p\left(w\right) - alpha_p\left(w\right)beta_p\left(v\right)$
for v, w ∈ TpM. (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

$alpha wedge \left(beta + gamma\right) = alpha wedge beta + alpha wedge gamma.$
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 k-form and β is an l-form, then
$alpha wedge beta = \left(-1\right)^\left\{kl\right\} beta wedge alpha.$

Exterior differential complex

One important property of the exterior derivative is that d2 = 0. This means that the exterior derivative defines a cochain complex:

$0 toOmega^0\left(M\right) stackrel\left\{mathrm d\right\}\left\{to\right\} Omega^1\left(M\right) stackrel\left\{mathrm d\right\}\left\{to\right\} Omega^2\left(M\right) stackrel\left\{mathrm d\right\}\left\{to\right\} Omega^3\left(M\right) to cdots$

where i is the inclusion of constant functions. By the Poincaré lemma, this complex is locally exact except at Ω0(M). Its cohomology is the de Rham cohomology of M.

On a Riemannian manifold, additional operations, such as the Hodge star operator and codifferential can be defined.

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 : MN is smooth and ω is a smooth k-form 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* : TMTN. Fix a differential k-form ω on N. For a point p of M and tangent vectors v1, ..., vk to M at p, the pullback of ω is defined by the formula

$\left(f^*omega\right)_p\left(v_1, ldots, v_k\right) = omega_\left\{f\left(p\right)\right\}\left(f_*v_1, ldots, f_*v_k\right).$
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
$M stackrel\left\{f\right\}\left\{to\right\} N stackrel\left\{omega\right\}\left\{to\right\} T^*M stackrel\left\{\left(Df\right)^*\right\}\left\{longrightarrow\right\} T^*M.$

Pullback respects all of the basic operations on forms:

$f^*\left(omega + eta\right) = f^*omega + f^*eta,$
$f^*\left(omegawedgeeta\right) = f^*omegawedge f^*eta,$
$f^*\left(mathrm\left\{d\right\}omega\right) = mathrm\left\{d\right\}\left(f^*omega\right).$

The pullback of a form can also be written in coordinates. Assume that x1, ..., xm are coordinates on M, that y1, ..., yn are coordinates on N, and that these coordinate systems are related by the formulas yi = fi(x1, ..., xm) for all i. Then, locally on N, ω can be written as

$omega = sum_\left\{i_1 < cdots < i_k\right\} omega_\left\{i_1cdots i_k\right\}mathrm\left\{d\right\}y_\left\{i_1\right\} wedge cdots wedge mathrm\left\{d\right\}y_\left\{i_k\right\},$
where, for each choice of i1, ..., ik, $omega_\left\{i_1cdots i_k\right\}$ is a real-valued function of y1, ..., yn. Using the linearity of pullback and its compatibility with wedge product, the pullback of ω has the formula
$f^*omega = sum_\left\{i_1 < cdots < i_k\right\} \left(omega_\left\{i_1cdots i_k\right\}circ f\right)mathrm\left\{d\right\}f_\left\{i_1\right\}wedgecdotswedgemathrm\left\{d\right\}f_\left\{i_n\right\}.$
Each exterior derivative dfi can be expanded in terms of dx1, ..., dxm. The resulting k-form can be written using Jacobian matrices:
$f^*omega = sum_\left\{i_1 < cdots < i_k\right\} sum_\left\{j_1 < cdots < j_k\right\} \left(omega_\left\{i_1cdots i_k\right\}circ f\right)frac\left\{partial\left(f_\left\{i_1\right\}, ldots, f_\left\{i_k\right\}\right)\right\}\left\{partial\left(x_\left\{j_1\right\}, ldots, x_\left\{j_k\right\}\right)\right\}mathrm\left\{d\right\}x_\left\{j_1\right\} wedge cdots wedgemathrm\left\{d\right\}x_\left\{j_k\right\}.$

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.

Let

$omega=sum a_\left\{i_1,dots,i_k\right\}\left(\left\{mathbf x\right\}\right),\left\{mathrm d\right\}x^\left\{i_1\right\} wedge cdots wedge \left\{mathrm d\right\}x^\left\{i_k\right\}$

be a differential form and S a differentiable k-manifold over which we wish to integrate, where S has the parameterization

$S\left(\left\{mathbf u\right\}\right)=\left(x^1\left(\left\{mathbf u\right\}\right),dots,x^n\left(\left\{mathbf u\right\}\right)\right)$

for u in the parameter domain D. Then [Rudin, 1976] defines the integral of the differential form over S as

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

where

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

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

The fundamental relationship between the exterior derivative and integration is given by the general Stokes' theorem: If $omega$ is an n−1-form with compact support on M and ∂M denotes the boundary of M with its induced orientation, then

$int_M mathrm \left\{d\right\}omega = oint_\left\{partial M\right\} omega.!,$

This theorem also underlies the duality between de Rham cohomology and the homology of chains.

Applications 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

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

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

$textbf\left\{J\right\} = J^a epsilon_\left\{abcd\right\}, \left\{mathrm d\right\}x^b wedge \left\{mathrm d\right\}x^c wedge \left\{mathrm d\right\}x^d.$

Using these definitions, Maxwell's equations can be written very compactly in geometrized units as

$mathrm\left\{d\right\}, \left\{textbf\left\{F\right\}\right\} = textbf\left\{0\right\}$
$mathrm\left\{d\right\}, \left\{*textbf\left\{F\right\}\right\} = textbf\left\{J\right\}$

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

The 2-form $* mathbf\left\{F\right\}$ is also called Maxwell 2-form.

Applications 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.