Differential Form

Differential Form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to defining 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 from one-variable calculus is called a 1-form, and can be integrated over an interval in the domain of f:

and similarly the expression: f(x,y,z) dx∧dy + g(x,y,z) dx∧dz + h(x,y,z) dy∧dz is a 2-form that has a surface integral over an oriented surface S:

Likewise, a 3-form f(x, y, z) dx∧dy∧dz represents something that can be integrated over a region of space.

The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration. There is an operation d on differential forms known as the exterior derivative that, when acting on a k-form produces a (k+1)-form. This operation extends the differential of a function, and the divergence and the curl of a vector field in an appropriate sense that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the general Stokes' theorem. In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as De Rham's theorem.

The general setting for the study of differential forms is on a differentiable manifold. Differential 1-forms are naturally dual to vector fields on a 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 the information is expressed in terms of differential forms. As a particular example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.