Välisvõrrandid

Välisvõrrandid, also known as exterior algebra or exterior calculus, is a branch of mathematics that generalizes the concept of the cross product in Euclidean space. It deals with multilinear alternating forms, which are functions that take multiple vectors as input and return a scalar, and change sign if any two input vectors are swapped. The core concept is the wedge product, denoted by the symbol $\wedge$. The wedge product of two vectors produces a bivector, which can be thought of as representing a plane segment.

The key property of the wedge product is its anticommutativity: $u \wedge v = -v \wedge u$. This implies that if two vectors are linearly dependent, their wedge product is zero. This is analogous to how the cross product of parallel vectors in 3D is zero. Välisvõrrandid allows for the construction of higher-order objects called k-vectors, which represent k-dimensional oriented subspaces.

This algebraic structure has profound applications in differential geometry and physics. It provides a natural framework for defining differential forms, which are fundamental to the calculus of manifolds. The exterior derivative, an operation on differential forms, generalizes the gradient, divergence, and curl of vector calculus. The integration of differential forms over higher-dimensional surfaces leads to powerful theorems like Stokes' theorem, which unifies several classical theorems of vector calculus, including the divergence theorem and Green's theorem. Välisvõrrandid provides a coordinate-independent way to express geometric and physical laws.