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Element of an exterior algebra
In multilinear algebra, a multivector, sometimes called Clifford number or multor, is an element of the exterior algebra Λ(V) of a vector space V. This
Multivector
Infinitesimal calculus on functions defined on a geometric algebra
to a general multivector, called the multivector derivative. Let F {\displaystyle F} be a multivector-valued function of a multivector. The directional
Geometric_calculus
Algebraic structure designed for geometry
Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric
Geometric_algebra
In differential geometry, a field in mathematics, a multivector field, polyvector field of degree k {\displaystyle k} , or k {\displaystyle k} -vector
Polyvector_field
Object in geometric algebra
Cleven(V) consisting of multivectors R such that R R ~ = 1. {\displaystyle R{\tilde {R}}=1.} That is, it consists of multivectors that can be written as
Rotor_(mathematics)
Branch of mathematics
respect to each argument. It involves concepts such as matrices, tensors, multivectors, systems of linear equations, higher-dimensional spaces, determinants
Multilinear_algebra
as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields
Schouten–Nijenhuis_bracket
Broad concept generalizing scalars in mathematics and physics
historical reasons. Vector quaternion, a quaternion with a zero real part Multivector or p-vector, an element of the exterior algebra of a vector space. Spinors
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Algebra associated to any vector space
while a more general sum of blades of arbitrary degree is called a multivector. The linear span of the k {\displaystyle k} -blades is called the k {\displaystyle
Exterior_algebra
Exterior algebraic map taking tensors from p forms to n-p forms
G b {\displaystyle a^{\mathrm {T} }Gb} , where a and b are arbitrary multivectors represented by 2 n × 1 {\displaystyle 2^{n}\times 1} column matrices
Hodge_star_operator
Algebraic object with geometric applications
in three dimensions e.g. Riemann curvature tensor 2 bivector, e.g. Poisson structure, inverse metric tensor e.g. elasticity tensor ⋮ N multivector ⋮
Tensor
Mapping from p forms to p-1 forms
a vector X (which has grade 1), a homogeneous multivector a having grade p, and an arbitrary multivector b, the right interior product satisfies the rule
Interior_product
Array of numbers
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Matrix_(mathematics)
Concept in linear algebra
generalized to the notions of projection and rejection of a general multivector onto/from any invertible k-blade. Scalar projection Vector notation Perwass
Vector_projection
Mathematical function, in linear algebra
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Linear_map
Topics referred to by the same term
and physics, k-vector may refer to: A wave vector k Crystal momentum A multivector of grade k, also called a k-vector, the dual of a differential k-form
K-vector
Physical quantity that changes sign with improper rotation
terminology to describe the various combinations is provided. For example, a multivector is a summation of k-fold wedge products of various k-values. A k-fold
Pseudovector
Mathematical operation on vectors in 3D space
product could be generalised to arbitrary multivectors in three dimensions, which results in a multivector consisting of only elements of grades 1 (1-vectors/true
Cross_product
Matrix operation which flips a matrix over its diagonal
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Transpose
Algebraic operation on coordinate vectors
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Dot_product
Provides integral formulas for all derivatives of a holomorphic function
{\displaystyle f(r)} can, in principle, be composed of any combination of multivectors. The proof of Cauchy's integral theorem for higher dimensional spaces
Cauchy's_integral_formula
Shorthand notation for tensor operations
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Einstein_notation
Expression that may be integrated over a region
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Differential_form
Branch of mathematics
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Differential_geometry
Branch of physics which studies the behavior of materials modeled as continuous media
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Continuum_mechanics
Property of a mathematical space
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Dimension
Topics referred to by the same term
syndrome, a rare liver disorder Rotor (mathematics), an even-graded multivector used to produce rotations and some other affine transformations Curl
Rotor
Topological space that locally resembles Euclidean space
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Manifold
Setting of relativistic physics in geometric algebra
products of vectors and then reversing the order of each product. For multivector A {\textstyle A} arising from a product of vectors, a 1 a 2 … a r − 1
Spacetime_algebra
Straight path on a curved surface or a Riemannian manifold
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Geodesic
Four-dimensional number system
{\displaystyle \operatorname {Cl} _{3,0}(\mathbb {R} ).} This is an associative multivector algebra built up from fundamental basis elements σ1, σ2, σ3 using the
Quaternion
Set of vectors used to define coordinates
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Basis_(linear_algebra)
American physicist and science educator
mathematics and physics (Kluwer: Dordrecht/Boston, 1984). D. Hestenes, Multivector Calculus, Journal of Mathematical Analysis and Applications 24: 313–325
David_Hestenes
Mathematical operation on vector spaces
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensor_product
1989–1991 mass movement in Ukraine
Liberation Revolution in Ukraine as a product of a social movement and multivector network of structures]. Kharkiv National University of Internal Affairs
1989–1991 Ukrainian revolution
1989–1991_Ukrainian_revolution
Theory of gravitation as curved spacetime
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
General_relativity
Vector behavior under coordinate changes
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Method for specifying point positions
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Coordinate_system
Index of articles associated with the same name
scalar plus a bivector. The geometric product is well defined for any multivectors as arguments. A bilinear product in an algebra over a field. A Lie bracket
Vector_multiplication
Non-tensorial representation of the spin group
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Spinor
Type of derivative in differential geometry
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Lie_derivative
Mathematical object that describes the electromagnetic field in spacetime
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Electromagnetic_tensor
finite-dimensional geometric algebras (GA). The elements of UGA are called multivectors. Every multivector can be written as the sum of several r-vectors. Some r-vectors
Universal_geometric_algebra
Continuous surjection satisfying a local triviality condition
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Fiber_bundle
Tensor in differential geometry
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Ricci_curvature
Quaternion of norm 1 (unit quaternion)
{\displaystyle {\mathcal {G}}(3,0)} . In a departure from Hamilton's definition, multivector versors are not required to have unit norm, just to be invertible. Normalisation
Versor
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Kronecker_delta
Algebra of 4D spacetime
conjugation. If g ∈ G 3 {\displaystyle g\in \mathbb {G} _{3}} is an arbitrary multivector and ⟨ g ⟩ j {\displaystyle \langle g\rangle _{j}} projects g {\displaystyle
Algebra_of_physical_space
Mathematical Concept
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Voigt_notation
Conserved physical quantity; rotational analogue of linear momentum
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Angular_momentum
extension of linear algebra building upon concepts of p-vectors and multivectors with Grassmann algebra. Multiplicative number theory a subfield of analytic
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
economic ambitions. At the heart of its international diplomacy is a multivector foreign policy, which aims to maintain balanced and diverse relations
Foreign relations of Kazakhstan
Foreign_relations_of_Kazakhstan
Bilateral relations
relationship, however, is limited by historical memory and Vietnam's multivector foreign policy. While fears about regime change have lowered, the U.S
United States–Vietnam relations
United_States–Vietnam_relations
Antisymmetric permutation object acting on tensors
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Levi-Civita_symbol
Element of a unital algebra over the field of real numbers
\\0&i\not =j.\end{cases}}} Imposing closure under multiplication generates a multivector space spanned by a basis of 2 k {\displaystyle 2^{k}} elements, { 1
Hypercomplex_number
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Moment_of_inertia
Specification of a derivative along a tangent vector of a manifold
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Covariant_derivative
Universal construction in multilinear algebra
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensor_algebra
Operation in mathematics
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensor_contraction
Tensor field in Riemannian geometry
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Riemann_curvature_tensor
Tensor describing energy momentum density in spacetime
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Stress–energy_tensor
Study of curves from a differential point of view
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Differentiable_curve
Abbreviation in the fields of special and general relativity
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Four-tensor
Theory of interwoven space and time by Albert Einstein
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Special_relativity
Assignment of a tensor continuously varying across a region of space
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensor_field
Measure of the curvature of a pseudo-Riemannian manifold
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Weyl_tensor
Operation that pairs a left and a right R-module into an abelian group
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensor_product_of_modules
Mathematical notation
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Multi-index_notation
Differential form of degree one or section of a cotangent bundle
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
One-form
Ternary operation on vectors
ISBN 0-521-00551-5. Janne Pesonen. "Geometric Algebra of One and Many Multivector Variables" (PDF). p. 37. "Permutation Tensor". Wolfram. Retrieved 21
Triple_product
Affine connection on the tangent bundle of a manifold
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Levi-Civita_connection
Isomorphism between the tangent and cotangent bundles of a manifold
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Musical_isomorphism
Topics referred to by the same term
differential form with a vector field Left contraction and right contraction of multivectors in a geometric algebra, extensions of the inner product One of the rules
Contraction
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensors in curvilinear coordinates
Tensors_in_curvilinear_coordinates
Array of numbers describing a metric connection
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Christoffel_symbols
Second order tensor in vector algebra
and polyadic. Kronecker product Bivector Polyadic algebra Unit vector Multivector Differential form Quaternions Field (mathematics) The cross product only
Dyadics
Society becoming more democratic
Liberation Revolution in Ukraine as a product of a social movement and multivector network of structures]. Kharkiv National University of Internal Affairs
Democratization
Formulations of electromagnetism
0}(\mathbb {R} )} , the field and current are represented by multivectors. The field multivector, known as the Riemann–Silberstein vector, is F = E + I c
Mathematical descriptions of the electromagnetic field
Mathematical_descriptions_of_the_electromagnetic_field
differential forms on a Poisson manifold form a Gerstenhaber algebra. The multivector fields on a manifold form a Gerstenhaber algebra using the Schouten–Nijenhuis
Gerstenhaber_algebra
Structure defining distance on a manifold
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Metric_tensor
Tensor that describes the 4D geometry of spacetime
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Tensor used in general relativity
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Einstein_tensor
Operation on differential forms
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Exterior_derivative
Unital algebra homomorphism of exterior algebras
between vector spaces is a natural extension of the map to arbitrary multivectors. It is the unique unital algebra homomorphism of exterior algebras whose
Outermorphism
Representation of mechanical stress at every point within a deformed 3D object
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Cauchy_stress_tensor
Decomposition in multilinear algebra
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensor_rank_decomposition
Coordinate-free definition of a tensor
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensor_(intrinsic_definition)
Differential form
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Volume_form
System of moving vectors in differential geometry
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Parallel_transport
Tensor index notation for tensor-based calculations
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Ricci_calculus
Object in differential geometry
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Torsion_tensor
Mathematics of smooth surfaces
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Differential geometry of surfaces
Differential_geometry_of_surfaces
Ways of writing certain laws of physics
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Tensor operator generalizes the notion of operators which are scalars and vectors
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensor_operator
Generalization of tensor fields
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Tensor_density
Electromagnetism in general relativity
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Tensor invariant under permutations of vectors it acts on
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Symmetric_tensor
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Symmetrization
Sum of a scalar and vector in Clifford algebra
defined as the inversion of the sign of odd-grade multivectors, while maintaining the even-grade multivectors invariant: Four special subspaces can be defined
Paravector
Topics referred to by the same term
Schouten–Nijenhuis bracket (2 versions, defined on either symmetric or antisymmetric multivectors) This disambiguation page lists articles associated with the title Nijenhuis
Nijenhuis_bracket
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR
Girl/Female
Indian, Sanskrit
Victorious in the End
Boy/Male
English, Hindu, Indian
Victory
Girl/Female
Tamil
Vallika | வலà¯à®²à®¿à®•à®¾
Diamond, Creeper
Girl/Female
Indian, Modern
Love
Girl/Female
Australian, Greek, Spanish
Bitter; Sea of Bitterness; The Perfect One; Sea of Sorrow; Wished for Child; Rebellious; Star; Similar to Maria
Girl/Female
Hindu, Indian
Wife of Pandavs
Girl/Female
Latin Russian American
Light.
Girl/Female
Indian
Goddess Durga
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Full of Rays
Boy/Male
Irish
Noble chief.
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR