1 Answers
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 dx is an example of a 1-form, and can be integrated over an interval contained in the domain of f:
Similarly, the expression f dx ∧ dy + g dz ∧ dx + h dy ∧ dz is a 2-form that can be integrated over a surface S:
The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f 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 d x , d y , … . {\displaystyle dx,dy,\ldots.} On an n-dimensional manifold, the top-dimensional form is called a volume form.