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 is an example of a -form, and can be integrated over an interval contained in the domain of :
Similarly, the expression is a -form that can be integrated over a surface :
The symbol denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials On an -dimensional manifold, the top-dimensional form (-form) is called a volume form.
The differential forms form an alternating algebra. This implies that and This alternating property reflects the orientation of the domain of integration.
The exterior derivative is an operation on differential forms that, given a -form , produces a -form This operation extends the differential of a function (a function can be considered as a -form, and its differential is ) 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 -forms are naturally dual to vector fields on a differentiable manifold, and the pairing between vector fields and -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. 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 -form can be integrated over an oriented manifold of dimension . A differential -form can be thought of as measuring an infinitesimal oriented length, or 1-dimensional oriented density. A differential -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 , and negatively oriented otherwise. If then the integral of the differential -form over the interval (with its natural positive orientation) is
which is the negative of the integral of the same differential form over the same interval, when equipped with the opposite orientation. That is:
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 (), the increment is negative in the direction of integration.
More generally, an -form is an oriented density that can be integrated over an -dimensional oriented manifold. (For example, a -form can be integrated over an oriented curve, a -form can be integrated over an oriented surface, etc.) If is an oriented -dimensional manifold, and is the same manifold with opposite orientation and is an -form, then one has: