NationMaster - Encyclopedia: Exterior calculus

In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V over a field K is a certain unital associative algebra which contains V as a subspace. It is denoted by Λ(V) or Λ^•(V) and its multiplication, known as the wedge product or the exterior product, is written as ∧. The wedge product is associative and bilinear; its essential property is that it is alternating on V: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In mathematics, associativity is a property that a binary operation can have. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

Note that these three properties are only valid for the vectors in V, not for all elements of the algebra Λ(V). In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

The exterior algebra is in fact the "most general" algebra with these properties. This means that all equations that hold in the exterior algebra follow from the above properties alone. This generality of Λ(V) is formally expressed by a certain universal property, see below. In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

Elements of the form v₁∧v₂∧…∧v_k with v₁,…,v_k in V are called k-vectors. The subspace of Λ(V) generated by all k-vectors is known as the k-th exterior power of V and denoted by Λ^k(V). The exterior algebra can be written as the direct sum of each of the k-th powers: In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

The exterior product has the important property that the product of a k-vector and an l-vector is a k+l-vector. Thus the exterior algebra forms a graded algebra where the grade is given by k. These k-vectors have geometric interpretations: the 2-vector u∧v represents the oriented parallelogram with sides u and v, while the 3-vector u∧v∧w represents the oriented parallelepiped with edges u, v, and w. In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... A parallelogram. ... In geometry, a parallelepiped or parallelopipedon is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ...

Exterior powers find their main application in differential geometry, where they are used to define differential forms. As a consequence, there is a natural wedge product for differential forms. All of these concepts go back to Hermann Grassmann. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Hermann Günther Grassmann (April 15, 1809 – September 26, 1877) was a German mathematician, physicist, linguist, scholar, and neohumanist. ...

Basis and dimension

If the dimension of V is n and {e₁,...,e_n} is a basis of V, then the set In mathematics, the dimension of a vector space V is the cardinality (i. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

is a basis for the k-th exterior power Λ^k(V). The reason is the following: given any wedge product of the form

then every vector v_j can be written as a linear combination of the basis vectors e_i; using the bilinearity of the wedge product, this can be expanded to a linear combination of wedge products of those basis vectors. Any wedge product in which the same basis vector appears more than once is zero; any wedge product in which the basis vectors don't 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 v_j in terms of the basis e_i. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ... For the square matrix section, see square matrix. ...

Counting the basis elements, we see that the dimension of Λ^k(V) is n choose k. In particular, Λ^k(V) = {0} for k > n. In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ...

The exterior algebra is a graded algebra as the direct sum In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a grading. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

(where we set Λ⁰(V) = K and Λ¹(V) = V), and therefore its dimension is equal to the sum of the binomial coefficients, which is 2ⁿ.

Example: the exterior algebra of Euclidean 3-space

For vectors in R³, the exterior algebra is closely related to the cross product and triple product. Using the standard basis {i, j, k}, the wedge product of a pair of vectors In mathematics, the cross product is a binary operation on vectors in a three dimensional vector space. ... This article is about mathematics. ...

where {i ∧ j, i ∧ k, j ∧ k} is the basis for the three-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 vectors in a three dimensional vector space. ...

where i ∧ j ∧ k is the basis vector for the one-space Λ³(R³). This imitates the usual definition of the triple product. This article is about mathematics. ...

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 v and w. 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 linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ... In mathematics, an orthonormal basis of an inner product space V(i. ...

The space Λ¹(R³) is R³, and the space Λ⁰(R³) is R. Direct-summing all four subspaces together yields a vector space Λ(R³) of eight-dimensional vectors

Then given a pair of eight-dimensional vectors a and b, with a given as above and

It is easy to verify by inspection that the eight-dimensional wedge product has the vector (1,0,0,0,0,0,0,0) as the multiplicative unit element. It is also possible to verify by multiplying out components that this Λ(R³) algebra wedge product is associative (as well as bilinear):

Universal property and construction

Let V be a vector space over the field K (which in most applications will be the field of real numbers). The fact that Λ(V) is the "most general" unital associative K-algebra containing V with an alternating multiplication on V can be expressed formally by the following universal property: In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

Given any unital associative K-algebra A and any K-linear map j : V → A 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 mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... A homomorphism between two algebras over a field K, A and B, is a map such that for all k in K and x,y in A, F(kx)=kF(x) F(x+y)=F(x)+F(y) F(xy)=F(x)F(y) Categories: Math stubs | Algebra ...

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. We thus take the two-sided ideal I in T(V) generated by all elements of the form v⊗v for v in V, and define Λ(V) as the quotient In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, the term ideal has multiple meanings. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ...

(and use ∧ as the symbol for multiplication in Λ(V)). It is then straightforward to show that Λ(V) contains V and satisfies the above universal property.

Rather than defining Λ(V) first and then identifying the exterior powers Λ^k(V) as certain subspaces, one may alternatively define the spaces Λ^k(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.

Anti-symmetric operators and exterior powers

Given two vector spaces V and X, an anti-symmetric operator from V^k to X is a multilinear map In linear algebra, a multilinear map is a mathematical function of several vector variables that is linear in each variable. ...

such that whenever v₁,...,v_k are linearly dependent vectors in V, then In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

The most famous example is the determinant, an anti-symmetric operator from (Kⁿ)ⁿ to K. In linear algebra, a determinant is a function depending on n that associates a scalar det(A) to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

which associates to k vectors from V their wedge product, i.e. their corresponding k-vector, is also anti-symmetric. In fact, this map is the "most general" anti-symmetric operator defined on V^k: given any other anti-symmetric operator f : V^k → X, there exists a unique linear map φ: Λ^k(V) → X with f = φ o w. This universal property characterizes the space Λ^k(V) and can serve as its definition. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

The set of all anti-symmetric maps from V^k to the base field K is a vector space, as the sum of two such maps, or the multiplication of such a map with a scalar, is again anti-symmetric. If V has finite dimension n, then this space can be identified with Λ^k(V^∗), where V^∗ denotes the dual space of V. In particular, the dimension of the space of anti-symmetric maps from V^k to K is n choose k. In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ...

Under this identification, and if the base field is R or C, the wedge product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose ω : V^k → K and η : V^m → K 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. It is defined as follows: In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ...

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: In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

The interior product

If V* denotes the dual space to the vector space V, then for each $alphain V^*$ , it is possible to define a derivation on the algebra $bigwedge V$ , In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1Ã—n) and column vectors (nÃ—1). ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

Suppose that ${bold w}inbigwedge^k V$ . Then w is a multilinear mapping of V* to R, so it is defined by its values on the k-fold Cartesian product $V^*times V^*timesdotstimes V^*$ . If $u_0,u_1,dots,u_{k-2}$ are k-1 elements of V*, then we define In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ...

where in each term of the summation, "

α i

" occupies the i-th position among the arguments of w. Additionally, we require that

(i α f) = 0

whenever f is a pure scalar (i.e., belonging to

Λ 0 V

Alternatively in index notation, if ${bold w}=w_{i_0i_1dots i_k}$ is a skew-symmetric k form in $bigwedge^kV$ , then $i_alpha{bold w}$ is a skew-symmetric k-1 form in $bigwedge^{k-1}V$ given by In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...

Axiom 1. For each k and each $alphain V^*$ , there exists a graded derivation of degree -1 :

Axiom 2. If v is an element of V, then $i_{alpha}v=langle alpha,v rangle$ is the dual pairing between elements of V and elements of V*.

Index notation

In the index notation, used primarily by physicists, Index notation is used in mathematics to refer to the elements of matrices or the components of a vector. ... A physicist is a scientist trained in physics. ...

$(omegawedgeeta)_{a_1 cdots a_{k+m}}=frac{1}{k!m!}epsilon_{a_1 cdots a_{k+m}}^{b_1 cdots b_k c_1 cdots c_m} omega_{b_1 cdots b_k} eta_{c_1 cdots c_m}$

Differential forms

Let M be a differentiable manifold. A differential k-form ω is a section of Λ^kT^∗M, the k-th exterior power of the cotangent bundle of M. Equivalently, ω is a smooth function on M which assigns to each point x of M an element of Λ^k(T_xM)^∗. Roughly speaking, differential forms are globalized versions of cotangent vectors. Differential forms are important tools in differential geometry, where, among other things, they are used to define de Rham cohomology and Alexander-Spanier cohomology. This page is about a higher mathematics topic. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ... In mathematics, particularly in algebraic topology Alexander_Spanier cohomology is a cohomology theory arising from differential forms with compact support on a manifold. ...

Generalization

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). It will satisfy the analogous universal property. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ...

Physical applications

Grassmann algebras have some important applications in physics where they are used to model various concepts related to fermions and supersymmetry. Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

See also: superspace, superalgebra, supergroup Superspace has had two meanings in physics. ... In mathematics and theoretical physics, a superalgebra over a field K generally refers to a Z2-graded algebra over K (here Z2 is the cyclic group of order 2). ... The concept of supergroup is a generalization of a that of group. ...

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