Exterior algebra

The exterior algebra of a vector space is an algebra generated by its vectors under an alternating product called the wedge product. Its elements encode antisymmetric combinations and include differential forms and oriented volume elements.

Wedge product: The bilinear product in an exterior algebra that satisfies v ∧ v = 0 for every vector v. It is the defining operation that makes vector products alternating.
Antisymmetry: A property in which exchanging two arguments changes the sign of a quantity. Swapping vectors in a wedge product reverses its sign.
Differential form: A smooth field of alternating multilinear functions on tangent vectors of a manifold. Differential forms take values in exterior powers of cotangent spaces.
Oriented volume: A volume measure that changes sign when orientation is reversed. Top-degree wedge products encode oriented volume elements.
Alternating multilinear map: A multilinear map that vanishes whenever two of its arguments are equal. The wedge product encodes alternating multilinear behavior.
Graded-commutative algebra: A graded algebra whose homogeneous elements commute up to a sign determined by their degrees. Wedge products obey this signed commutation rule.
Exterior derivative: An operator that maps differential forms to forms of one higher degree and satisfies d² = 0. It uses the wedge product to build coordinate-independent differentiation.
Linear independence: A property of vectors for which no nontrivial linear combination equals zero. A wedge of vectors is nonzero exactly when those vectors are linearly independent.
Graded algebra: An algebra decomposed into components indexed by degree, with multiplication combining degrees. Exterior powers form the degree components, and wedge products add their degrees.
Determinant: A scalar associated with a square matrix that measures signed volume scaling and detects singularity. The coefficient of a top-degree wedge product is a determinant.

Linked from 9 pages