välisvõrrandite

Välisvõrrandid, also known as exterior differential forms or differential forms, are fundamental objects in differential geometry and multilinear algebra. They are functions that take one or more vectors from a tangent space at a point on a manifold and return a scalar value. Specifically, a k-form assigns a multilinear, alternating map to each tangent space. The "alternating" property means that if two input vectors are swapped, the output is negated, and if two input vectors are the same, the output is zero. This property is crucial for their geometric interpretation.

The set of all k-forms at a point forms a vector space, and the collection of all k-forms on a manifold forms a differential algebra. Operations on exterior forms include the exterior product (wedge product), denoted by '∧', which combines two forms to create a higher-order form, and the exterior derivative, denoted by 'd'. The exterior derivative is a generalization of the gradient, curl, and divergence operators from vector calculus. For instance, the exterior derivative of a 0-form (a function) is a 1-form (its gradient), the exterior derivative of a 1-form is a 2-form (related to curl), and the exterior derivative of a 2-form is a 3-form (related to divergence). A key property is that the exterior derivative is nilpotent, meaning that applying it twice always results in zero (d² = 0). Exterior forms are essential for formulating generalizations of Stokes' theorem, which relate integrals over regions to integrals over their boundaries. They play a significant role in physics, particularly in electromagnetism and general relativity, as well as in topology and analysis.