In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, 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 contained in the domain of f:

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

Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that can be integrated over a surface S:

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

The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ 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 ${\displaystyle dx,dy,\ldots .}$ On an n-dimensional manifold, the top-dimensional form (n-form) is called a volume form.

The differential forms form an alternating algebra. This implies that ${\displaystyle dy\wedge dx=-dx\wedge dy}$ and ${\displaystyle dx\wedge dx=0.}$ 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 ${\displaystyle \varphi }$, produces a (k+1)-form ${\displaystyle d\varphi .}$ This operation extends the differential of a function (a function can be considered as a 0-form, and its differential is ${\displaystyle df(x)=f'(x)dx}$). 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.

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 with reference to his 1899 paper.^[1] 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).

Concept

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

Integration and orientation

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 , 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 (with its natural positive orientation) is

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

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

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

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:

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

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 ${\textstyle \int _{A}f\,d\mu =\int _{}f\,d\mu }$ 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, dx¹, ..., dxⁿ 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:

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

where the f_k = f_k(x¹, ... , xⁿ) are functions of all the coordinates. A differential 1-form is integrated along an oriented curve as a line integral.

The expressions dxⁱ ∧ 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 xⁱ–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 the exterior product (the symbol is the wedge ∧). This is similar to the cross product from vector calculus, in that it is an alternating product. For instance,

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

because the square whose first side is dx¹ and second side is dx² is to be regarded as having the opposite orientation as the square whose first side is dx² and whose second side is dx¹. This is why we only need to sum over expressions dxⁱ ∧ dx^j, with i < j; for example: a(dxⁱ ∧ dx^j) + b(dx^j ∧ dxⁱ) = (a − b) dxⁱ ∧ dx^j. The exterior product allows higher-degree differential forms to be built out of lower-degree ones, in much the same way that the cross product in vector calculus allows one to compute the area vector of a parallelogram from vectors pointing up the two sides. Alternating also implies that dxⁱ ∧ dxⁱ = 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^i₁ ∧ ⋅⋅⋅ ∧ dx^i_m = 0 if any two of the indices i₁, ..., i_m are equal, in the same way that the "volume" enclosed by a parallelotope whose edge vectors are linearly dependent is zero.

Multi-index notation

A common notation for the wedge product of elementary k-forms is so called multi-index notation: in an n-dimensional context, for ${\displaystyle I=(i_{1},i_{2},\ldots ,i_{k}),1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}$, 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 Zdroj:https://en.wikipedia.org?pojem=Exterior_calculus
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