Differential form

Uncommon card · No. 134467

In mathematics, differential forms provide a unified approach to define 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 is an example of a 1-form, and can be integrated over an interval [a,b] contained in the domain of f: ∫abf(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: ∫S(f(x,y,z)dx∧dy+g(x,y,z)dz∧dx+h(x,y,z)dy∧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 dx,dy,…. On an n-dimensional...

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