Differential form

In mathematics, differential forms provide a unified approach to define 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 \(f(x) \, dx\) is an example of a 1-form, and can be integrated over an interval \([a,b]\) contained in the domain of \(f\): \[\int_a^b f(x)\,dx.\] Similarly, the expression \[f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\] is a 2-form that can be integrated over a surface \(S\): \[\int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right).\] The symbol \(\wedge\) denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form \(f(x,y,z) \, dx \wedge dy \wedge dz\) represents a volume element that 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 is homogeneous of degree \(k\) in the coordinate differentials \(dx, dy, \ldots.\) On an \(n\)-dimensional manifold, a top-dimensional form (\(n\)-form) is called a volume form.

The differential forms form an alternating algebra. This implies that \(dy \wedge dx = -dx \wedge dy\) and \(dx \wedge dx = 0.\) This alternating property reflects the orientation of the domain of integration.

The exterior derivative is an operation on differential forms that, given a \(k\)-form \(\varphi\), produces a \((k+1)\)-form \(d\varphi.\) This operation extends the differential of a function (a function can be considered as a \(0\)-form, and its differential is \(df(x) = f'(x) \, dx\)). This allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general result, the generalized Stokes theorem.

Differential \(1\)-forms are naturally dual to vector fields on a differentiable manifold, and the pairing between vector fields and \(1\)-forms 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 that 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.

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 with reference to his 1899 paper. Some aspects of the exterior algebra of differential forms appears in Hermann Grassmann's 1844 work, Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).

A differential k-form can be integrated over an oriented manifold of 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 on oriented 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 the conventions for 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 a chain. In measure 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 \(\int_A f\,d\mu = \int_{[a,b]} f\,d\mu\) to indicate integration over a subset A. This is a minor distinction in one dimension, but becomes subtler on higher-dimensional manifolds; see below for details.

Making the notion of an oriented density precise, and thus of a differential form, involves the exterior algebra. The differentials of a set of coordinates, dx, ..., dx can be used as a basis for all 1-forms. Each of these represents a covector at 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, ... , x) are functions of all the coordinates. A differential 1-form is integrated along an oriented curve as a line integral.

The expressions dx ∧ dx, 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–x-plane. A general 2-form is a linear combination of these at every point on the manifold: \(\sum_{1 \leq i

Condensed: the full section is in Wikipedia.

A common notation for the wedge product of elementary \(k\)-forms is so called multi-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 \(dx^I := dx^{i_1} \wedge \cdots \wedge dx^{i_k} = \bigwedge_{i\in I} dx^i\). 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

In addition to the exterior product, there is also the exterior derivative operator \(d\). The exterior derivative of a differential form is a generalization of the differential of a function, in the sense that the exterior derivative of \(f\in C^{\infty}(M)=\Omega^0(M)\) is exactly the differential of \(f\). When generalized to higher forms, if \(\omega=f\text{d}x^I\) is a simple \(k\)-form, then its exterior derivative \(d\omega\) 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 \(\tau = \sum_{I \in \mathcal{J}_{k,n}} a_I \, dx^I \in \Omega^k(M)\), 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 \(\mathbb{R}^3\), with the Hodge star operator, the exterior derivative corresponds to gradient, curl, and divergence, 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 in differential 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 an antiderivation of degree 1 on the exterior algebra of differential forms. The benefit of this more general approach is that it allows for a natural coordinate-free approach to integrate on manifolds. It also allows for a natural generalization of the fundamental theorem of calculus, called the (generalized) Stokes' theorem, which is a central result in the theory of integration on manifolds.

Let \(U\) be an open set in \(\mathbb{R}^n\). A differential \(0\)-form ("zero-form") is defined to be a smooth function \(f\) on \(U\), the set of which is denoted \(C^{\infty}(U)\). If \(\mathbf{v}\) is any vector in \(\mathbb{R}^n\), then \(f\) has a directional derivative \(\partial_{\mathbf{v}}f\), 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 \(\mathbf{v}\) 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 \(\mathbf{v}\) is a vector field on \(U\) by evaluating \(\mathbf{v}\) 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 vector, i.e., ∂f / ∂x, where x, x, ..., x are the coordinate vectors in U. By their very definition, partial derivatives depend upon the choice of coordinates: if new coordinates y, y, ..., y 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 ∂_v f (p) is a linear function of v:

\[\begin{align} (\partial_{\mathbf{v} + \mathbf{w}} f)(p) &= (\partial_\mathbf{v} f)(p) + (\partial_\mathbf{w} f)(p) \\ (\partial_{c \mathbf{v}} f)(p) &= c (\partial_\mathbf{v} f)(p) \end{align}\]

for any vectors v, w and any real number c. At each point p, this linear map from R to R is denoted df_p and called the derivative or differential of f at p. Thus df_p(v) = ∂_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 a linear combination Σ ve_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, x, ..., x are themselves functions on U, and so define differential 1-forms dx, dx, ..., dx. Let f = x. Since ∂x / ∂x = 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 \[df_p = \sum_{i=1}^n \frac{\partial f}{\partial x^i}(p) (dx^i)_p .\] Applying both sides to e_j, the result on each side is the jth partial derivative of f at p. Since p and j were arbitrary, this proves the formula (*).

Condensed: the full section is in Wikipedia.

Let \(M\) be a smooth manifold. A smooth differential form of degree \(k\) is a smooth section of the \(k\)th exterior power of the cotangent bundle of \(M\). The set of all differential \(k\)-forms on a manifold \(M\) is a vector space, often denoted \(\Omega^k(M)\).

The definition of a differential form may be restated as follows. At any point \(p\in M\), a \(k\)-form \(\beta\) defines an element \[\beta_p \in {\textstyle\bigwedge}^k T_p^* M,\] where \(T_pM\) is the tangent space to \(M\) at \(p\) and \(T^*_p(M)\) is its dual space. This space is naturally isomorphic to the fiber at \(p\) of the dual bundle of the \(k\)th exterior power of the tangent bundle of \(M\). That is, \(\beta\) is also a linear functional \(\beta_p \colon {\textstyle\bigwedge}^k T_pM \to \mathbf{R}\), 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 an alternating 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 \(\alpha\) assigns to each point \(p\in M\) a linear functional \(\alpha_p\) on \(T_pM\). In the presence of an inner product on \(T_pM\) (induced by a Riemannian metric on \(M\)), \(\alpha_p\) may be represented as the inner product with a tangent vector \(X_p\). Differential \(1\)-forms are sometimes called covariant 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 \(p\), \[\operatorname{Alt}(\tau_p)(x_1, \dots, x_k) = \frac{1}{k!}\sum_{\sigma \in S_k} \sgn(\sigma) \tau_p(x_{\sigma(1)}, \dots, x_{\sigma(k)}),\] where \(S_k\) is the symmetric group on \(k\) 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 \(\beta\) as a totally antisymmetric covariant tensor field of rank \(k\). The differential forms on \(M\) are in one-to-one correspondence with such tensor fields.

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 the exterior product of two differential forms, the exterior derivative of a single differential form, the interior product of a differential form and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with respect to a vector field on a manifold with a defined connection.

The exterior product of a \(k\)-form \(\alpha\) and an \(\ell\)-form \(\beta\), denoted \(\alpha\wedge\beta\), is a \((k+\ell)\)-form. At each point \(p\) of the manifold \(M\), the forms \(\alpha\) and \(\beta\) 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 \(\alpha\wedge\beta\) 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 \(\alpha\otimes\beta\) 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 \(k=\ell=1\), then \(\alpha\wedge\beta\) is the \(2\)-form whose value at a point \(p\) is the alternating bilinear form defined by \[(\alpha\wedge\beta)_p(v,w)=\alpha_p(v)\beta_p(w) - \alpha_p(w)\beta_p(v)\] for \(v,w\in T_pM\).

The exterior product is bilinear: If \(\alpha\), \(\beta\), and \(\gamma\) 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 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 \(\alpha\) is a \(k\)-form and \(\beta\) is an \(\ell\)-form, then \[\alpha \wedge \beta = (-1)^{k\ell} \beta \wedge \alpha .\] One also has the graded Leibniz rule: \[d(\alpha\wedge\beta)=d\alpha\wedge\beta + (-1)^{k}\alpha\wedge d\beta.\]

On a Riemannian manifold, or more generally a pseudo-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 the Hodge star operator \(\star \colon \Omega^k(M)\ \stackrel{\sim}{\to}\ \Omega^{n-k}(M)\) and the codifferential \(\delta\colon \Omega^k(M)\rightarrow \Omega^{k-1}(M)\), which has degree −1 and is adjoint to the exterior differential d.

One important property of the exterior derivative is that \(d^2=0\). This means that the exterior derivative defines a cochain 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 its cohomology is by definition the de Rham cohomology of \(M\). By the Poincaré lemma, the de Rham complex is locally exact except at \(\Omega^0(M)\). The kernel at \(\Omega^0(M)\) is the space of locally constant functions on \(M\). Therefore, the complex is a resolution of the constant sheaf \(\underline{\mathbf{R}}\), which in turn implies a form of de Rham's theorem: de Rham cohomology computes the sheaf cohomology of \(\underline{\mathbf{R}}\).

Suppose that f : M → N is smooth. The differential of f is a smooth map df : TM → TN between the tangent bundles of M and N. This map is also denoted f_∗ and called the pushforward. For any point p ∈ M and any tangent vector v ∈ T_pM, 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 ∈ N has two or more preimages, then the vector field may determine two or more distinct vectors in T_qN. If f is not surjective, then there will be a point q ∈ 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 → 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 → N be smooth, and let ω be a smooth k-form on N. Then there is a differential form fω on M, called the pullback of 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 v₁, ..., 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 TN of N. Using to denote a dual map, the dual to the differential of f is (df) : TN → TM. 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 \(\bigwedge^k (df)^*\) denote the kth 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 of vector bundles \[{\textstyle\bigwedge}^k TN\ \stackrel{\omega}{\to}\ N \times \mathbf{R},\] where N × 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 × R. The vector bundle morphism \({\textstyle\bigwedge}^k TM \to M \times \mathbf{R}\) defined in this way is fω.

Condensed: the full section is in Wikipedia.

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.

Let \(U\) be an open subset of \(\mathbb{R}^n\). Give \(\mathbb{R}^n\) its standard orientation and \(U\) the restriction of that orientation. Every smooth \(n\)-form \(\omega\) on \(U\) has the form \[\omega = f(x)\,dx^1 \wedge \cdots \wedge dx^n\] for some smooth function \(f:\mathbb{R}^n\rightarrow\mathbb{R}\). Such a function has an integral in the usual Riemann or Lebesgue sense. This allows us to define the integral of \(\omega\) 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, \(\text{d}x^1\wedge\text{d}x^2\) must be the negative of the integral of \(\text{d}x^2\wedge\text{d}x^1\). 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.