Differential form

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 [a, b] 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.