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In mathematics, the wedge product, also known as exterior product, is an anti-symmetrisation (alternation) of the tensor product. The wedge product is a distributive associative multiplication of skew-symmetric multilinear maps which is anti-commutative for maps with odd number of variables and commutative otherwise. The systematic theory starts from the exterior power construction for a vector space.

As in the case of tensor products, the number of variables of the wedge of two maps is the sum of the numbers of their variables:

Definition:

where k and m are the numbers of variables for each of the two skew-symmetric functions and alternation of a map is defined to be the signed average of the values over all the permutations of its variables:

The wedge product makes pointwise sense for differential forms.

## Wedge product of spaces, exterior powers

The wedge product of two vector spaces may be identified with the subspace of their tensor product generated by the skew-symmetric tensors. (This definition, though, works only over fields of characteristic zero. In algebraic work one may need an alternate definition, based on a universal property. This means taking an appropriate quotient of the tensor product, instead - of the same dimension. The difference is harmless for real and complex vector spaces.)

The wedge product of a vector space V with itself k times is called its k-th exterior power and is denoted . If dim V=n, then dim is n-choose-k.

Example: Let be the dual space of V, i.e. space of all linear maps from V to R. The second exterior power is the space of all skew-symmetric bilinear maps from VxV to R.

## Definition in generality

The definition of an anti-symmetric multilinear operator is an operator m: Vn -> X such that if there is a linear dependence between its arguments, the result is 0. Note that the addition of anti-symmetric operators, or multiplying one by a scalar, is still anti-symmetric -- so the anti-symmetric multilinear operators on Vn form a vector space.

The most famous example of an anti-symmetric operator is the determinant.

The nth wedge space W, for a module V over a commutative ring R, together with the anti-symmetric linear wedge operator w: Vn -> W is such that for every n-linear anti-symmetric operator m: Vn -> X there exists a unique linear operator l: W -> X such that m = l o w. The wedge is unique up to a unique isomorphism.

One way of defining the wedge space constructively is by dividing the Tensor space by the subspace generated by all the tensors of n-tuples which are linearily dependent.

The dimension of the kth wedge space for a free module of dimension n is n! / (k!(n-k)!). In particular, that means that up to a constant, there is a single anti-symmetric functional with the arity of the dimension of the space. Also note that every linear functional is anti-symmetric.

Note that the wedge operator commutes with the * operator. In other words, we can define a wedge on functionals such that the result is an anti-symmetric multilinear functional. In general, we can define the wedge of an n-linear anti-symmetric functional and an m-linear anti-symmetric functional to be an (n+m)-linear anti-symmetric functional. Since it turns out that this operation is associative, we can also define the power of an anti-symmetric linear functional.

When dealing with differentiable manifolds, we define an "n-form to be a function from the manifold to the n-th wedge of the cotangent bundle. Such a form will be said to be differentiable if, when applied to n differentiable vector fields, the result is a differentiable function.  