Differential forms

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 an example of a 1-form, and can be integrated over an interval [a, b] 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:

In the above expressions, and in the balance of this article, 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 k-dimensional sets, and is homogeneous of degree k in the coordinate differentials.

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