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Alternating Form
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—''parallelepiped'' and ''cube'' in three dimensions, ''parallelogram'' and ''square'' in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only ''parallelograms'' and ''parallelepipeds'' exist. Three equivalent definitions of ''parallelepiped'' are *a polyhedron with six faces ( hexahedron), each of which is a parallelogram, *a hexahedron with three pairs of parallel faces, and *a prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped. "Parallelepiped" is now usually pronounced ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (geometry)
In geometry, the orientation, angular position, attitude, bearing, or direction of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it occupies. More specifically, it refers to the imaginary rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement. It may be necessary to add an imaginary translation, called the object's location (or position, or linear position). The location and orientation together fully describe how the object is placed in space. The above-mentioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, and its location does not change when it rotates. Euler's rotation theorem shows that in three dimensions any orientation can be reached with a single rotation around a fixed axis. This gives one common way of representi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded \Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' joins tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-vector
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a representation space of the standard representation of the Lorentz group, the (,) representation. It differs from a Euclidean vector in how its magnitude is determined. The transformations that preserve this magnitude are the Lorentz transformations, which include spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). Four-vectors describe, for instance, position in spacetime modeled as Minkowski space, a particle's four-momentum , the amplitude of the electromagnetic four-potential at a point in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra. The Lorentz group may be represented by 4×4 matrices . The action of a Lorentz transformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivector
In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (also known as decomposable -vectors or -blades) of the form : v_1\wedge\cdots\wedge v_k, where v_1, \ldots, v_k are in . A -vector is such a linear combination that is ''homogeneous'' of degree (all terms are -blades for the same ). Depending on the authors, a "multivector" may be either a -vector or any element of the exterior algebra (any linear combination of -blades with potentially differing values of ). In differential geometry, a -vector is a vector in the exterior algebra of the tangent vector space; that is, it is an antisymmetric tensor obtained by taking linear combinations of the exterior product of tangent vectors, for some integer . A differential -form is a -vector in the exterior algebra of the dual of the tangent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Triple Product
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. Scalar triple product The scalar triple product (also called the mixed product, box product, or triple scalar product) is defined as the dot product of one of the vectors with the cross product of the other two. Geometric interpretation Geometrically, the scalar triple product : \mathbf\cdot(\mathbf\times \mathbf) is the (signed) volume of the parallelepiped defined by the three vectors given. Here, the parentheses may be omitted without causing ambiguity, since the dot product cannot be evaluated first. If it were, it would leave the cross product of a scalar and a vector, which is not defined. Properties * The scalar triple product is unchanged under a circular shift of its three operands (a, b, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
