In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which Like the cross product, and the scalar triple product, the exterior product of vectors is used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs. This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically In linear algebra, the exterior product provides an abstract algebraic basis-independent manner for describing the determinant and the minors of a linear transformation, and is fundamentally related to ideas of rank and linear independence. Linear algebra is the branch of Mathematics concerned with Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Linear algebra, a minor of a matrix A is the Determinant of some smaller Square matrix, cut down from A by removing In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that The column rank of a matrix A is the maximal number of Linearly independent columns of A. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors

The exterior algebra (also known as the Grassmann algebra, after Hermann Grassmann[1]) of a given vector space V over a field K is the algebra generated by the exterior product. Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an algebra over a field K, or a K -algebra, is a Vector space A over K equipped with It is widely used in contemporary geometry, especially differential geometry and algebraic geometry through the algebra of differential forms, as well as in multilinear algebra and related fields. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Mathematics, multilinear algebra extends the methods of Linear algebra.

Formally, the exterior algebra is a certain unital associative algebra over the field K, containing V as a subspace. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. It is denoted by Λ(V) or Λ(V) and its multiplication is also known as the wedge product or the exterior product and is written as $\wedge$. The wedge product is an associative and bilinear operation:

$\wedge: \Lambda(V) \times \Lambda(V) \to\Lambda(V).$
$(\alpha,\beta)\mapsto \alpha\wedge\beta.$

Its essential feature is that it is alternating on V:

(1) $v\wedge v = 0 \mbox{ for all }v\in V,$

which implies in particular

(2) $u\wedge v = - v\wedge u$ for all $u,v\in V$, and
(3) $v_1\wedge v_2\wedge\cdots \wedge v_k = 0$ whenever $v_1, \ldots, v_k \in V$ are linearly dependent. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, a bilinear map is a function of two arguments that is linear in each [2]

In terms of category theory, the exterior algebra is a type of functor on vector spaces, given by a universal construction. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism The universal construction allows the exterior algebra to be defined, not just for vector spaces over a field, but also for modules over a commutative ring, and for other structures of interest. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the wedge product. In Mathematics, a bialgebra over a field K is a structure which is both a Unital Associative algebra and a Coalgebra over In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals This dual algebra is precisely the algebra of alternating multilinear forms on V, and the pairing between the exterior algebra and its dual is given by the interior product. In Multilinear algebra, a multilinear form is a map of the type f V^N \to K where V is a Vector space In Mathematics, the interior product is a degree &minus1 derivation on the Exterior algebra of Differential forms on a Smooth manifold

## Motivating examples

### Areas in the plane

The area of a parallelogram in terms of the determinant of the matrix of coordinates of two of its vertices.

The Cartesian plane R2 is a vector space equipped with a basis consisting of a pair of unit vectors

${\mathbf e}_1 = (1,0),\quad {\mathbf e}_2 = (0,1).$

Suppose that

${\mathbf v} = v_1{\mathbf e}_1 + v_2{\mathbf e}_2, \quad {\mathbf w} = w_1{\mathbf e}_1 + w_2{\mathbf e}_2$

are a pair of given vectors in R2, written in components. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length There is a unique parallelogram having v and w as two of its sides. The area of this parallelogram is given by the standard determinant formula:

$A = \left|\det\begin{bmatrix}{\mathbf v}& {\mathbf w}\end{bmatrix}\right| = |v_1w_2 - v_2w_1|.$

Consider now the exterior product of v and w:

${\mathbf v}\wedge {\mathbf w} = (v_1{\mathbf e}_1 + v_2{\mathbf e}_2)\wedge (w_1{\mathbf e}_1 + w_2{\mathbf e}_2)=v_1w_1{\mathbf e}_1\wedge{\mathbf e}_1+ v_1w_2{\mathbf e}_1\wedge {\mathbf e}_2+v_2w_1{\mathbf e}_2\wedge {\mathbf e}_1+v_2w_2{\mathbf e}_2\wedge {\mathbf e}_2$
$=(v_1w_2-v_2w_1){\mathbf e}_1\wedge{\mathbf e}_2$

where the first step uses the distributive law for the wedge product, and the last uses the fact that the wedge product is alternating. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n Note that the coefficient in this last expression is precisely the determinant of the matrix [v w]. The fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the signed area of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation.

The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if A(v,w) denotes the signed area of the parallelogram determined by the pair of vectors v and w, then A must satisfy the following properties:

1. A(av,bw) = a b A(v,w) for any real numbers a and b, since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram).
2. A(v,v) = 0, since the area of the degenerate parallelogram determined by v (i. for the degeneracy of a Graph, see Arboricity#Related_concepts. e. , a line segment) is zero. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points
3. A(w,v) = -A(v,w), since interchanging the roles of v and w reverses the orientation of the parallelogram.
4. A(v + aw,w) = A(v,w), since adding a multiple of w to v affects neither the base nor the height of the parallelogram and consequently preserves its area.
5. A(e1, e2) = 1, since the area of the unit square is one.

With the exception of the last property, the wedge product satisfies the same formal properties as the area. In a certain sense, the wedge product generalizes the final property by allowing the area of a parallelogram to be compared to that of any "standard" chosen parallelogram. In other words, the exterior product in two-dimensions is a basis-independent formulation of area. [3]

### Cross and triple products

For vectors in R3, the exterior algebra is closely related to the cross product and triple product. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion. Using the standard basis {e1, e2, e3}, the wedge product of a pair of vectors

$\mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3$

and

$\mathbf{v} = v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2 + v_3 \mathbf{e}_3$

is

$\mathbf{u} \wedge \mathbf{v} = (u_1 v_2 - u_2 v_1) (\mathbf{e}_1 \wedge \mathbf{e}_2) + (u_1 v_3 - u_3 v_1) (\mathbf{e}_1 \wedge \mathbf{e}_3) + (u_2 v_3 - u_3 v_2) (\mathbf{e}_2 \wedge \mathbf{e}_3)$

where {e1 Λ e2, e1 Λ e3, e2 Λ e3} is the basis for the three-dimensional space Λ2(R3). This imitates the usual definition of the cross product of vectors in three dimensions. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

Bringing in a third vector

$\mathbf{w} = w_1 \mathbf{e}_1 + w_2 \mathbf{e}_2 + w_3 \mathbf{e}_3,$

the wedge product of three vectors is

$\mathbf{u} \wedge \mathbf{v} \wedge \mathbf{w} = (u_1 v_2 w_3 + u_2 v_3 w_1 + u_3 v_1 w_2 - u_1 v_3 w_2 - u_2 v_1 w_3 - u_3 v_2 w_1) (\mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3)$

where e1 Λ e2 Λ e3 is the basis vector for the one-dimensional space Λ3(R3). This imitates the usual definition of the triple product. This article is about mathematics See Lawson criterion for the use of the term triple product in relation to Nuclear fusion.

The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product u×v can be interpreted as a vector which is perpendicular to both u and v and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the minors of the matrix with columns u and v. In Linear algebra, a minor of a matrix A is the Determinant of some smaller Square matrix, cut down from A by removing The triple product of u, v, and w is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns u, v, and w. The exterior product in three-dimensions allows for similar interpretations. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions. In Mathematics, an orthonormal basis of an Inner product space V (i

## Formal definitions and algebraic properties

The exterior algebra Λ(V) over a vector space V is defined as the quotient algebra of the tensor algebra by the two-sided ideal I generated by all elements of the form $x \otimes x$ such that xV. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra [4] Symbolically,

$\Lambda(V) := T(V)/I.\,$

The wedge product ∧ of two elements of Λ(V) is defined by

$\alpha\wedge\beta = \alpha\otimes\beta \pmod I.$

### Anticommutativity of the wedge product

This product is anticommutative on elements of V, for supposing that x, yV,

$0 \equiv (x+y)\wedge (x+y) = x\wedge x + x\wedge y + y\wedge x + y\wedge y \equiv x\wedge y + y\wedge x \pmod I$

whence

$x\wedge y = - y\wedge x.$

More generally, if x1x2, . In mathematics anticommutativity refers to the property of an operation being anticommutative, i . . , xk are elements of V, and σ is a permutation of the integers [1,. In Mathematics, a permutation group is a group G whose elements are Permutations of a given set M, and whose group operation . . ,k], then

$x_{\sigma(1)}\wedge x_{\sigma(2)}\wedge\dots\wedge x_{\sigma(k)} = {\rm sgn}(\sigma)x_1\wedge x_2\wedge\dots \wedge x_k,$

where sgn(σ) is the signature of the permutation σ. In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even [5]

### The exterior power

The kth exterior power of V, denoted Λk(V), is the vector subspace of Λ(V) spanned by elements of the form

$x_1\wedge x_2\wedge\dots\wedge x_k,\quad x_i\in V, i=1,2,\dots, k.$

If α ∈ Λk(V), then α is said to be a k-multivector. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector If, furthermore, α can be expressed as a wedge product of k elements of V, then α is said to be decomposable. Although decomposable multivectors span Λk(V), not every element of Λk(V) is decomposable. For example, in R4, the following 2-multivector is not decomposable:

$\alpha = e_1\wedge e_2 + e_3\wedge e_4.$

(This is in fact a symplectic form, since α ∧ α ≠ 0. In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the [6])

#### Basis and dimension

If the dimension of V is n and {e1,. In Mathematics, the dimension of a Vector space V is the cardinality (i . . ,en} is a basis of V, then the set

$\{e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_k} \mid 1\le i_1 < i_2 < \cdots < i_k \le n\}$

is a basis for Λk(V). Basis vector redirects here For basis vector in the context of crystals see Crystal structure. The reason is the following: given any wedge product of the form

$v_1\wedge\cdots\wedge v_k$

then every vector vj can be written as a linear combination of the basis vectors ei; using the bilinearity of the wedge product, this can be expanded to a linear combination of wedge products of those basis vectors. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics Any wedge product in which the same basis vector appears more than once is zero; any wedge product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis k-vectors can be computed as the minors of the matrix that describes the vectors vj in terms of the basis ei. In Linear algebra, a minor of a matrix A is the Determinant of some smaller Square matrix, cut down from A by removing In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally

By counting the basis elements, the dimension of Λk(V) is the binomial coefficient n choose k. In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x   k term in the Polynomial In particular, Λk(V) = {0} for k > n.

Any element of the exterior algebra can be written as a sum of multivectors. Hence, as a vector space the exterior algebra is a direct sum

$\Lambda(V) = \Lambda^0(V)\oplus \Lambda^1(V) \oplus \Lambda^2(V) \oplus \cdots \oplus \Lambda^n(V)$

(where by convention Λ0(V) = K and Λ1(V) = V), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2n. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

#### Rank of a multivector

If α ∈ Λk(V), then it is possible to express α as a linear combination of decomposable multivectors:

$\alpha = \alpha^{(1)} + \alpha^{(2)} + \cdots + \alpha^{(s)}$

where each α(i) is decomposable, say

$\alpha^{(i)} = \alpha^{(i)}_1\wedge\cdots\wedge\alpha^{(i)}_k,\quad i=1,2,\dots, s.$

The rank of the multivector α is the minimal number of decomposable multivectors in such an expansion of α. This is similar to the notion of tensor rank. In Mathematics, the modern Component-free approach to the theory of Tensors views tensors initially as Abstract objects expressing some definite type of

Rank is particularly important in the study of 2-multivectors (Sternberg 1974, §III. 6) (Bryant et al. 1991). The rank of a 2-multivector α can be identified with the rank of the matrix of coefficients of α in a basis. The column rank of a matrix A is the maximal number of Linearly independent columns of A. Thus if ei is a basis for V, then α can be expressed uniquely as

$\alpha = \sum_{i,j}a_{ij}e_i\wedge e_j$

where aij = −aji (the matrix of coefficients is skew-symmetric). The rank of α agrees with the rank of the matrix aij.

In characteristic 0, the 2-multivector α has rank p if and only if

$\underset{p}{\underbrace{\alpha\wedge\cdots\wedge\alpha}}\not= 0$

and

$\underset{p+1}{\underbrace{\alpha\wedge\cdots\wedge\alpha}} = 0.$

### Graded structure

The wedge product of a k-multivector with a p-multivector is a (k+p)-multivector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section

$\Lambda(V) = \Lambda^0(V)\oplus \Lambda^1(V) \oplus \Lambda^2(V) \oplus \cdots \oplus \Lambda^n(V)$

gives the exterior algebra the additional structure of a graded algebra. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure Symbolically,

$\left(\Lambda^k(V)\right)\wedge\left(\Lambda^p(V)\right)\sub \Lambda^{k+p}(V).$

Moreover, the wedge product is graded anticommutative, meaning that if α ∈ Λk(V) and β ∈ Λp(V), then

$\alpha\wedge\beta = (-1)^{kp}\beta\wedge\alpha.$

In addition to studying the graded structure on the exterior algebra, Bourbaki (1989) studies additional graded structures on exterior algebras, such as those on the exterior algebra of a graded module (a module that already carries its own gradation). In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure

### Universal property

Let V be a vector space over the field K. Informally, multiplication in Λ(V) is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identities vv = 0 for vV and vw = -wv for v, wV. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Mathematics, associativity is a property that a Binary operation can have Formally, Λ(V) is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative K-algebra containing V with alternating multiplication on V must contain a homomorphic image of Λ(V). In other words, the exterior algebra has the following universal property:[7]

Given any unital associative K-algebra A and any K-linear map j : VA such that j(v)j(v) = 0 for every v in V, then there exists precisely one unital algebra homomorphism f : Λ(V) → A such that f(v) = j(v) for all v in V. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that A homomorphism between two algebras over a field K, A and B, is a map FA\rightarrow B such that for all k

To construct the most general algebra that contains V and whose multiplication is alternating on V, it is natural to start with the most general algebra that contains V, the tensor algebra T(V), and then enforce the alternating property by taking a suitable quotient. In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the We thus take the two-sided ideal I in T(V) generated by all elements of the form vv for v in V, and define Λ(V) as the quotient

Λ(V) = T(V)/I

(and use Λ as the symbol for multiplication in Λ(V)). In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. It is then straightforward to show that Λ(V) contains V and satisfies the above universal property.

As a consequence of this construction, the operation of assigning to a vector space V its exterior algebra Λ(V) is a functor from the category of vector spaces to the category of algebras. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships

Rather than defining Λ(V) first and then identifying the exterior powers Λk(V) as certain subspaces, one may alternatively define the spaces Λk(V) first and then combine them to form the algebra Λ(V). This approach is often used in differential geometry and is described in the next section.

### Generalizations

Given a commutative ring R and an R-module M, we can define the exterior algebra Λ(M) just as above, as a suitable quotient of the tensor algebra T(M). In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars It will satisfy the analogous universal property. Many of the properties of Λ(M) also require that M be a projective module. In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation Where finite-dimensionality is used, the properties further require that M be finitely generated and projective. Generalizations to the most common situations can be found in (Bourbaki 1989).

Exterior algebras of vector bundles are frequently considered in geometry and topology. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely-generated projective modules, by the Serre-Swan theorem. In the mathematical fields of Topology and K-theory, Swan's theorem, also called the Serre–Swan theorem, relates the geometric notion of More general exterior algebras can be defined for sheaves of modules. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

## Duality

### Alternating operators

Given two vector spaces V and X, an alternating operator (or anti-symmetric operator) from Vk to X is a multilinear map

f: VkX

such that whenever v1,. In Linear algebra, a multilinear map is a Mathematical function of several vector variables that is linear in each variable . . ,vk are linearly dependent vectors in V, then

f(v1,. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors . . ,vk) = 0.

The most famous example is the determinant, an alternating operator from (Kn)n to K. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

The map

w: Vk → Λk(V)

which associates to k vectors from V their wedge product, i. e. their corresponding k-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on Vk: given any other alternating operator f : VkX, there exists a unique linear map φ: Λk(V) → X with f = φ o w. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that This universal property characterizes the space Λk(V) and can serve as its definition. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism

### Alternating multilinear forms

The above discussion specializes to the case when X = K, the base field. In this case an alternating multilinear function

f : VkK

is called an alternating multilinear form. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the multiplication of such a map with a scalar, is again alternating. If V has finite dimension n, then this space can be identified with Λk(V), where V denotes the dual space of V. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In particular, the dimension of the space of anti-symmetric maps from Vk to K is the binomial coefficient n choose k. In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x   k term in the Polynomial

Under this identification, the wedge product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : VkK and η : VmK are two anti-symmetric maps. As in the case of tensor products of multilinear maps, the number of variables of their wedge product is the sum of the numbers of their variables. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector It is defined as follows:

$\omega\wedge\eta=\frac{(k+m)!}{k!\,m!}{\rm Alt}(\omega\otimes\eta)$

where the alternation Alt of a multilinear map is defined to be the signed average of the values over all the permutations of its variables:

${\rm Alt}(\omega)(x_1,\ldots,x_k)=\frac{1}{k!}\sum_{\sigma\in S_k}{\rm sgn}(\sigma)\,\omega(x_{\sigma(1)},\ldots,x_{\sigma(k)})$

This definition of the wedge product is well-defined even if the fields K has finite characteristic, if one considers an equivalent version of the above that does not use factorials or any constants:

$\omega \wedge \eta(x_1,\ldots,x_{k+m}) = \sum_{\sigma \in Sh_{k,m}} {\rm sgn}(\sigma)\,\omega(x_{\sigma(1)}, \ldots, x_{\sigma(k)}) \eta(x_{\sigma(k+1)}, \ldots, x_{\sigma(k+m)}),$

where here $Sh_{k,m} \subset S_{k+m}$ is the subset of k,m shuffles: permutations σ sending $1,2,\ldots,k$ to numbers $\sigma(1)< \sigma(2) < \cdots < \sigma(k)$, and $k+1,k+2,\ldots,k+m$ to numbers $\sigma(k+1)<\cdots<\sigma(k+m)$. In several fields of Mathematics the term permutation is used with different but closely related meanings In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's

(Note. Some conventions, particularly in physics, define the wedge product as

$\omega\wedge\eta={\rm Alt}(\omega\otimes\eta).$

This convention is not adopted here, but see the Alternating tensor algebra section below for further details. )

### Bialgebra structure

In formal terms, there is a correspondence between the graded dual of the graded algebra Λ(V) and alternating multilinear forms on V. The wedge product of multilinear forms defined above is dual to a coproduct defined on Λ(V), giving the structure of a coalgebra. In Category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological In Mathematics, coalgebras are structures that are dual to Unital Associative algebras The Axioms of unital associative algebras can

The coproduct is a linear function Δ : Λ(V) → Λ(V) ⊗ Λ(V) given on decomposable elements by

$\Delta(x_1\wedge\dots\wedge x_k) = \sum_{p=0}^k \sum_{\sigma\in Sh_{p,k-p}} {\rm sgn}(\sigma) (x_{\sigma(1)}\wedge\dots\wedge x_{\sigma(p)})\otimes (x_{\sigma(p+1)}\wedge\dots\wedge x_{\sigma(k)}).$

For example,

$\Delta(x_1) = 1 \otimes x_1 + x_1 \otimes 1,$
$\Delta(x_1 \wedge x_2) = 1 \otimes (x_1 \wedge x_2) + x_1 \otimes x_2 - x_2 \otimes x_1 + (x_1 \wedge x_2) \otimes 1.$

This extends by linearity to an operation defined on the whole exterior algebra. In terms of the coproduct, the wedge product on the dual space is just the graded dual of the coproduct:

$(\alpha\wedge\beta)(x_1\wedge\dots\wedge x_k) = (\alpha\otimes\beta)\left(\Delta(x_1\wedge\dots\wedge x_k)\right)$

where the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely, α∧β = ε o (α⊗β) o Δ, where ε is the counit, as defined presently).

The counit is the homomorphism ε : Λ(V) → K which returns the 0-graded component of its argument. The coproduct and counit, along with the wedge product, define the structure of a bialgebra on the exterior algebra. In Mathematics, a bialgebra over a field K is a structure which is both a Unital Associative algebra and a Coalgebra over

### The interior product

See also: interior product

Suppose that V is finite-dimensional. In Mathematics, the interior product is a degree &minus1 derivation on the Exterior algebra of Differential forms on a Smooth manifold If V* denotes the dual space to the vector space V, then for each α ∈ V*, it is possible to define an antiderivation on the algebra Λ(V),

$i_\alpha:\Lambda^k V\rightarrow\Lambda^{k-1}V.$

This derivation is called the interior product with α, or sometimes the insertion operator, or contraction by α. In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator

Suppose that w ∈ ΛkV. Then w is a multilinear mapping of V* to R, so it is defined by its values on the k-fold Cartesian product V*× V*× . Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. . . × V*. If u1, u2, . . . , uk-1 are k-1 elements of V*, then define

$(i_\alpha {\bold w})(u_1,u_2\dots,u_{k-1})={\bold w}(\alpha,u_1,u_2,\dots, u_{k-1}).$

Additionally, let iαf = 0 whenever f is a pure scalar (i. e. , belonging to Λ0V).

#### Axiomatic characterization and properties

The interior product satisfies the following properties:

1. For each k and each α ∈ V*,
$i_\alpha:\Lambda^kV\rightarrow \Lambda^{k-1}V.$
(By convention, Λ−1 = 0. )
2. If v is an element of V ( = Λ1V), then iαv = α(v) is the dual pairing between elements of V and elements of V*.
3. For each α ∈ V*, iα is a graded derivation of degree −1:
$i_\alpha (a\wedge b) = (i_\alpha a)\wedge b + (-1)^{\deg a}a\wedge (i_\alpha b).$

In fact, these three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case. In Mathematics, differential rings differential fields and differential algebras are rings, fields and algebras equipped with a derivation,

Further properties of the interior product include:

• $i_\alpha\circ i_\alpha = 0.$
• $i_\alpha\circ i_\beta = -i_\beta\circ i_\alpha.$

### Hodge duality

Main article: Hodge dual

Suppose that V has finite dimension n. In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W Then the interior product induces a canonical isomorphism of vector spaces

$\Lambda^k(V^*) \otimes \Lambda^n(V) \to \Lambda^{n-k}(V).$

In the geometrical setting, a non-zero element of the top exterior power Λn(V) (which is a one-dimensional vector space) is sometimes called a volume form (or orientation form, although this term may sometimes lead to ambiguity. In Mathematics, a volume form is a nowhere zero differential ''n''-form on an n - Manifold. ) Relative to a given volume form σ, the isomorphism is given explicitly by

$\alpha \in \Lambda^k(V^*) \mapsto i_\alpha\sigma \in \Lambda^{n-k}(V).$

If, in addition to a volume form, the vector space V is equipped with an inner product identifying V with V*, then the resulting isomorphism is called the Hodge dual (or more commonly the Hodge star operator)

$* : \Lambda^k(V) \rightarrow \Lambda^{n-k}(V).$

The composite of * with itself maps Λk(V) → Λk(V) and is always a scalar multiple of the identity map. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In most applications, the volume form is compatible with the inner product in the sense that it is a wedge product of an orthonormal basis of V. In Mathematics, an orthonormal basis of an Inner product space V (i In this case,

$*\circ * : \Lambda^k(V) \to \Lambda^k(V) = (-1)^{k(n-k) + q}I$

where I is the identity, and the inner product has metric signature (p,q) — p plusses and q minuses. The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive

## Functoriality

Suppose that V and W are a pair of vector spaces and f : VW is a linear transformation. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Then, by the universal construction, there exists a unique homomorphism of graded algebras

$\Lambda(f) : \Lambda(V)\rightarrow \Lambda(W)$

such that

$\Lambda(f)|_{\Lambda^1(V)} = f : V=\Lambda^1(V)\rightarrow W=\Lambda^1(W).$

In particular, Λ(f) preserves homogeneous degree. The k-graded components of Λ(f) are given on decomposable elements by

$\Lambda(f)(x_1\wedge \dots \wedge x_k) = f(x_1)\wedge\dots\wedge f(x_k).$

Let

$\Lambda^k(f) = \Lambda(f)_{\Lambda^k(V)} : \Lambda^k(V) \rightarrow \Lambda^k(W).$

The components of the transformation Λ(k) relative to a basis of V and W is the matrix of k × k minors of f. In particular, if V = W and V is of finite dimension n, then Λn(f) is a mapping of a one-dimensional vector space Λn to itself, and is therefore given by a scalar: the determinant of f. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

### Exactness

If

$0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0$

is a short exact sequence of vector spaces, then

$0\to \Lambda^1(U)\wedge\Lambda(V) \to \Lambda(V)\rightarrow \Lambda(W)\rightarrow 0$

is an exact sequence of graded vector spaces[8] as is

$0\to \Lambda(U)\to\Lambda(V).$[9]

### Direct sums

In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:

$\Lambda(V\oplus W)= \Lambda(V)\otimes\Lambda(W).$

This is a graded isomorphism; i. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group e. ,

$\Lambda^k(V\oplus W)= \bigoplus_{p+q=k} \Lambda^p(V)\otimes\Lambda^q(W).$

Slightly more generally, if

$0\rightarrow U\rightarrow V\rightarrow W\rightarrow 0$

is a short exact sequence of vector spaces then Λk(V) has a filtration

$0 = F^0 \subseteq F^1 \subseteq \dotsb \subseteq F^k \subseteq F^{k+1} = \Lambda^k(V)$

with quotients $F^{p+1}/F^p = \Lambda^{k-p}(U) \otimes \Lambda^p(W)$. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, a filtration is an Indexed set Si of Subobjects of a given Algebraic structure S, with the index In particular, if U is 1-dimensional then

$0\rightarrow U \otimes \Lambda^{k-1}(W) \rightarrow \Lambda^k(V)\rightarrow \Lambda^k(W)\rightarrow 0$

is exact, and if W is 1-dimensional then

$0\rightarrow \Lambda^k(U) \rightarrow \Lambda^k(V)\rightarrow \Lambda^{k-1}(U) \otimes W\rightarrow 0$

is exact. [10]

## The alternating tensor algebra

If K is a field of characteristic 0,[11] then the exterior algebra of a vector space V can be canonically identified with the vector subspace of T(V) consisting of antisymmetric tensors. In Mathematics and Theoretical physics, a Tensor is antisymmetric on two indices i and j if it flips sign when the two indices are Recall that the exterior algebra is the quotient of T(V) by the ideal I generated by xx.

Let Tr(V) be the space of homogeneous tensors of degree r. This is spanned by decomposable tensors

$v_1\otimes\dots\otimes v_r,\quad v_i\in V.$

The antisymmetrization (or sometimes the skew-symmetrization) of a decomposable tensor is defined by

$\text{Alt}(v_1\otimes\dots\otimes v_r) = \frac{1}{r!}\sum_{\sigma\in\mathfrak{S}_r} {\rm sgn}(\sigma) v_{\sigma(1)}\otimes\dots\otimes v_{\sigma(r)}$

where the sum is taken over the symmetric group of permutations on the symbols {1,. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying . . ,r}. This extends by linearity and homogeneity to an operation, also denoted by Alt, on the full tensor algebra T(V). The image Alt(T(V)) is the alternating tensor algebra, denoted A(V). This is a vector subspace of T(V), and it inherits the structure of a graded vector space from that on T(V). It carries an associative graded product $\widehat{\otimes}$ defined by

$t \widehat{\otimes} s = \text{Alt}(t\otimes s).$

Although this product differs from the tensor product, the kernel of Alt is precisely the ideal I (again, assuming that K has characteristic 0), and there is a canonical isomorphism

$A(V)\cong \Lambda(V).$

### Index notation

Suppose that V has finite dimension n, and that a basis e1, . . . , en of V is given. then any alternating tensor t ∈ Ar(V) ⊂ Tr(V) can be written in index notation as

$t = t^{i_1i_2\dots i_r}\, {\mathbf e}_{i_1}\otimes {\mathbf e}_{i_2}\otimes\dots\otimes {\mathbf e}_{i_r}$

where ti1 . Index notation is used in Mathematics to refer to the elements of matrices or the components of a vector. . .  ir is completely antisymmetric in its indices. In Mathematics and Theoretical physics, a Tensor is antisymmetric on two indices i and j if it flips sign when the two indices are

The wedge product of two alternating tensors t and s of ranks r and p is given by

$t\widehat{\otimes} s = \frac{1}{(r+p)!}\sum_{\sigma\in {\mathfrak S}_{r+p}}\text{sgn}(\sigma)t^{i_{\sigma(1)}\dots i_{\sigma(r)}}s^{i_{\sigma(r+1)}\dots i_{\sigma(r+p)}} {\mathbf e}_{i_1}\otimes {\mathbf e}_{i_2}\otimes\dots\otimes {\mathbf e}_{i_{r+p}}.$

The components of this tensor are precisely the skew part of the components of the tensor product st, denoted by square brackets on the indices:

$(t\widehat{\otimes} s)^{i_1\dots i_{r+p}} = t^{[i_1\dots i_r}s^{i_{r+1}\dots i_{r+p}]}.$

The interior product may also be described in index notation as follows. Let $t = t^{i_0i_1\dots i_{r-1}}$ be an antisymmetric tensor of rank r. Then, for α ∈ V*, iαt is an alternating tensor of rank r-1, given by

$(i_\alpha t)^{i_1\dots i_{r-1}}=r\sum_{j=0}^n\alpha_j t^{ji_1\dots i_{r-1}}.$

where n is the dimension of V.

## Applications

### Linear geometry

The decomposable k-vectors have geometric interpretations: the bivector $u\wedge v$ represents the plane spanned by the vectors, "weighted" with a number, given by the area of the oriented parallelogram with sides u and v. In Geometry, a parallelogram is a Quadrilateral with two sets of Parallel sides Analogously, the 3-vector $u\wedge v\wedge w$ represents the spanned 3-space weighted by the volume of the oriented parallelepiped with edges u, v, and w. Properties Any of the three pairs of parallel faces can be viewed as the base planes of the prism

### Projective geometry

Decomposable k-vectors in ΛkV correspond to weighted k-dimensional subspaces of V. Subspace may refer to;Mathematics Euclidean subspace, in linear algebra a set of vectors in n -dimensional Euclidean space that is closed under addition In particular, the Grassmannian of k-dimensional subspaces of V, denoted Grk(V), can be naturally identified with an algebraic subvariety of the projective space PkV). In Mathematics, a Grassmannian is a space which parameterizes all Linear subspaces of a Vector space V of a given Dimension. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which This is called the Plücker embedding. In the Mathematical fields of Algebraic geometry and Differential geometry (as well as Representation theory) the Plücker embedding describes

### Differential geometry

The exterior algebra has notable applications in differential geometry, where it is used to define differential forms. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is A differential form at a point of a differentiable manifold is an alternating multilinear form on the tangent space at the point. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since Equivalently, a differential form of degree k is a linear functional on the k-th exterior power of the tangent space. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional As a consequence, the wedge product of multilinear forms defines a natural wedge product for differential forms. Differential forms play a major role in diverse areas of differential geometry.

In particular, the exterior derivative gives the exterior algebra of differential forms on a manifold the structure of a differential algebra. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Mathematics, differential rings differential fields and differential algebras are rings, fields and algebras equipped with a derivation, The exterior derivative commutes with pullback along smooth mappings between manifolds, and it is therefore a natural differential operator. Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator The exterior algebra of differential forms, equipped with the exterior derivative, is a differential complex whose cohomology is called the de Rham cohomology of the underlying manifold and plays a vital role in the algebraic topology of differentiable manifolds. In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic

### Representation theory

In representation theory, the exterior algebra is one of the two fundamental Schur functors on the category of vector spaces, the other being the symmetric algebra. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a Young symmetrizer is an element of the group algebra of the Symmetric group, constructed in such a way that the image of the element In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field Together, these constructions are used to generate the irreducible representations of the general linear group. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation

### Physics

The exterior algebra is an archetypal example of a superalgebra, which plays a fundamental role in physical theories pertaining to fermions and supersymmetry. In Mathematics and Theoretical physics, a superalgebra is a Z 2- Graded algebra. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that For a physical discussion, see Grassmann number. In Mathematical physics, a Grassmann number (also called an anticommuting number or anticommuting C-number) is a mathematical construction which For various other applications of related ideas to physics, see superspace and supergroup (physics). " Superspace " has had two meanings in physics The word was first used by John Wheeler to describe the Configuration space of General relativity; for example The concept of supergroup is a Generalization of that of group.

## History

The exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension. Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a [12] This referred more generally to an algebraic (or axiomatic) theory of extended quantities and was one of the early precursors to the modern notion of a vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Saint-Venant also published similar ideas of exterior calculus for which he claimed priority over Grassmann. Adhémar Jean Claude Barré de Saint-Venant ( August 23, 1797 - January 1886 was a Mechanician and mathematician who contributed to early Stress analysis [13]

The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors. In a Grassmann algebra, a multivector is an element of a vector space V. It was thus a calculus, much like the propositional calculus, except focused exclusively on the task of formal reasoning in geometrical terms. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" [14] In particular, this new development allowed for an axiomatic characterization of dimension, a property that had previously only been examined from the coordinate point of view.

The import of this new theory of vectors and multivectors was lost to mid 19th century mathematicians,[15] until being thoroughly vetted by Giuseppe Peano in 1888. Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional Peano's work also remained somewhat obscure until the turn of the century, when the subject was unified by members of the French geometry school (notably Henri Poincaré, Elie Cartan, and Gaston Darboux) who applied Grassmann's ideas to the calculus of differential forms. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental Jean-Gaston Darboux ( August 14, 1842, Nîmes  &ndash February 23, 1917, Paris) was a French Mathematician In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is

A short while later, Alfred North Whitehead, borrowing from the ideas of Peano and Grassmann, introduced his universal algebra. Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" This then paved the way for the 20th century developments of abstract algebra by placing the axiomatic notion of an algebraic system on a firm logical footing. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules

## Notes

1. ^ Grassmann (1844) introduced these as extended algebras (cf. In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field In Mathematics, Clifford algebras are a type of Associative algebra. In Physics, quantization is a procedure for constructing a Quantum field theory starting from a classical field theory. In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Mathematics, multilinear algebra extends the methods of Linear algebra. In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra In Mathematical physics, a geometric algebra is a Multilinear algebra described technically as a Clifford algebra over a real vector space equipped In Mathematics, the Koszul complex was first introduced to define a Cohomology theory for Lie algebras by Jean-Louis Koszul (see Lie algebra Clifford 1878). He used the word äußere (literally translated as outer, or exterior) only to indicate the produkt he defined, which is nowadays conventionally called exterior product, probably to distinguish it from the outer product as defined in modern linear algebra. In Linear algebra, the outer product typically refers to the tensor product of two vectors. Linear algebra is the branch of Mathematics concerned with
2. ^ Note that, unlike associativity and bilinearity which are required for all elements of the algebra Λ(V), these last three properties are imposed only on the algebra's subspace V. The defining property (1) and property (3) are equivalent; properties (1) and (2) are equivalent unless the characteristic of K is two. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's
3. ^ This axiomatization of areas is due to Leopold Kronecker and Karl Weierstrass; see Bourbaki (1989, Historical Note). Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician For a modern treatment, see MacLane & Birkhoff (1999, Theorem IX. 2. 2). For an elementary treatment, see Strang (1993, Chapter 5).
4. ^ This definition is a standard one. See, for instance, MacLane & Birkhoff (1999).
5. ^ A proof of this can be found in more generality in Bourbaki (1989).
6. ^ See Sternberg (1964, §III. 6).
7. ^ See Bourbaki (1989, III. 7. 1), and MacLane & Birkhoff (1999, Theorem XVI. 6. 8). More detail on universal properties in general can be found in MacLane & Birkhoff (1999, Chapter VI), and throughout the works of Bourbaki.
8. ^ This part of the statement also holds in greater generality if V and W are modules over a commutative ring: That Λ converts epimorphisms to epimorphisms. See Bourbaki (1989, Proposition 3, III. 7. 2).
9. ^ This statement generalizes only to the case where V and W are projective modules over a commutative ring. Otherwise, it is generally not the case that Λ converts monomorphisms to monomorphisms. See Bourbaki (1989, Corollary to Proposition 12, III. 7. 9).
10. ^ Such a filtration also holds for vector bundles, and projective modules over a commutative ring. In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space This is thus more general than the result quoted above for direct sums, since not every short exact sequence splits in other abelian categories. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist
11. ^ See Bourbaki (1989, III. 7. 5) for generalizations.
12. ^ Kannenberg (2000) published a translation of Grassmann's work in English; he translated Ausdehnungslehre as Extension Theory.
13. ^ J Itard, Biography in Dictionary of Scientific Biography (New York 1970-1990).
14. ^ Authors have in the past referred to this calculus variously as the calculus of extension (Whitehead 1898; Forder 1941), or extensive algebra (Clifford 1878), and recently as extended vector algebra (Browne 2007).
15. ^ Bourbaki 1989, p.  661.

## References

### Mathematical references

• Bishop, R. & Goldberg, S. I. (1980), Tensor analysis on manifolds, Dover, ISBN 0-486-64039-6
Includes a treatment of alternating tensors and alternating forms, as well as a detailed discussion of Hodge duality from the perspective adopted in this article.
• Bourbaki, Nicolas (1989), Elements of mathematics, Algebra I, Springer-Verlag, ISBN 3-540-64243-9
This is the main mathematical reference for the article. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition It introduces the exterior algebra of a module over a commutative ring (although this article specializes primarily to the case when the ring is a field), including a discussion of the universal property, functoriality, duality, and the bialgebra structure. See chapters III. 7 and III. 11.
• Bryant, R. L. ; Chern, S.S.; Gardner, R. Shiing-Shen Chern (陳省身 pinyin: Chén Xǐngshēn October 26 1911 &ndash December 3 2004) was a Chinese American Mathematician B. ; Goldschmidt, H. L. & Griffiths, P.A. (1991), Exterior differential systems, Springer-Verlag
This book contains applications of exterior algebras to problems in partial differential equations. Phillip Griffiths (born 1938 is an American Mathematician, known for his work in the field of Geometry, and in particular for the Complex manifold approach In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Rank and related concepts are developed in the early chapters.
Chapter XVI sections 6-10 give a more elementary account of the exterior algebra, including duality, determinants and minors, and alternating forms. Saunders Mac Lane ( 4 August 1909, Taftville, Connecticut – 14 April 2005, San Francisco) was an American Garrett Birkhoff ( January 19, 1911, Princeton, New Jersey, USA – November
Contains a classical treatment of the exterior algebra as alternating tensors, and applications to differential geometry. Shlomo Zvi Sternberg is a leading Mathematician, known for his work in geometry particularly Symplectic geometry and the Differential geometry of

### Historical references

• Bourbaki, Nicolas (1989). "Historical note on chapters II and III", Elements of mathematics, Algebra I. Springer-Verlag.
• Clifford, W. (1878), “Applications of Grassmann's Extensive Algebra”, American Journal of Mathematics 1 (4): 350-358
• Forder, H. William Kingdon Clifford FRS ( May 4, 1845 &ndash March 3, 1879) was an English Mathematician and G. (1941), The Calculus of Extension, Cambridge University Press
• Grassmann, Hermann (1844), Die Lineale Ausdehnungslehre - Ein neuer Zweig der Mathematik  (The Linear Extension Theory - A new Branch of Mathematics)
• Kannenberg, Llyod (2000), Extension Theory (translation of Grassmann's Ausdehnungslehre), American Mathematical Society, ISBN 0821820311
• Peano, Giuseppe (1888), Calcolo Geometrico secondo l'Ausdehnungslehre di H. Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional Grassmann preceduto dalle Operazioni della Logica Deduttiva  [Geometric Calculus according to Grassmann's Ausdehnungslehre, preceded by the Operations of Deductive Logic]
• Whitehead, Alfred North (1898), A Treatise on Universal Algebra, with Applications, Cambridge

### Other references and further reading

• Browne, J. Alfred North Whitehead, OM ( February 15 1861, Ramsgate, Kent, England &ndash December 30 1947, M. (2007), Grassmann algebra - Exploring applications of Extended Vector Algebra with Mathematica, Published on line
An introduction to the exterior algebra, and geometric algebra, with a focus on applications. In Mathematical physics, a geometric algebra is a Multilinear algebra described technically as a Clifford algebra over a real vector space equipped Also includes a history section and bibliography.
• Spivak, Michael (1965), Calculus on manifolds, Addison-Wesley, ISBN 0-8053-90231-9
Includes applications of the exterior algebra to differential forms, specifically focused on integration and Stokes's theorem. Michael David Spivak (*1940 in Queens, New York) is a Mathematician specializing in Differential geometry, an expositor of Mathematics The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from The notation ΛkV in this text is used to mean the space of alternating k-forms on V; i. e. , for Spivak ΛkV is what this article would call ΛkV*. Spivak discusses this in Addendum 4.
• Strang, G. (1993), Introduction to linear algebra, Wellesley-Cambridge Press, ISBN 978-0961408855
Includes an elementary treatment of the axiomatization of determinants as signed areas, volumes, and higher-dimensional volumes. William Gilbert Strang, usually known as simply Gilbert Strang, is a renowned American Mathematician, with contributions to finite element theory
• Onishchik, A. L. (2001), “Exterior algebra”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
• Wendell H. The Encyclopaedia of Mathematics is a large reference work in Mathematics. Fleming (1965) Functions of Several Variables, Addison-Wesley. Addison-Wesley is a Book publishing imprint of Pearson PLC, best known for computer books
Chapter 6: Exterior algebra and differential calculus, pages 205-38. This textbook in multivariate calculus introduces the exterior algebra of differential forms adroitly into the calculus sequence for colleges. Multivariable calculus is the extension of Calculus in one Variable to calculus in several variables the functions which are differentiated and integrated involve

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