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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
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
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
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
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
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
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
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
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
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)
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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 Concept
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Voigt_notation
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
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
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
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
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
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
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)
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
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
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
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
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
Ways to represent 3D rotations
expected. This rotation formula is valid not only for vectors but for any multivector. In addition, when Euler angles are used, the complexity of the operation
Rotation formulations in three dimensions
Rotation_formulations_in_three_dimensions
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
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
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
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
Differential form
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Volume_form
Graphical notation for multilinear algebra calculations
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Penrose_graphical_notation
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
Vector satisfying some of the criteria of an eigenvector
algebra Symmetric algebra Clifford algebra Geometric algebra Bivector Multivector Gamas's theorem Affine and projective Affine space Affine transformation
Generalized_eigenvector
Construct allowing differentiation of tangent vector fields of manifolds
Fiber bundle Geodesic Levi-Civita connection Linear map Manifold Matrix Multivector Pseudotensor Spinor Vector Vector space Notable tensors Mathematics Kronecker
Affine_connection
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
Exterior product of vectors
Jacobian determinant of a change-of-coordinate function. Grassmannian Multivector Exterior algebra Differential form Geometric algebra Clifford algebra
Blade_(geometry)
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
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR
Male
English
Anglicized form of Hebrew Naphtaliy, NAFTALI means "my strife" or "wrestling."Â
Girl/Female
Hindu, Indian
Goddess Parvati / Laxmi / Saraswati
Girl/Female
Norse
Goddess of matrimonial love.
Boy/Male
Tamil
Srikanth | à®·à¯à®°à¯€Â கஂடÂ
Lover of wealth, Fortune, Respect, Esteem, Wisdom, Light
Boy/Male
Hindu, Indian
A Guru
Boy/Male
Muslim
Powerful, Strong, Ornamentation, Decoration
Boy/Male
Indian
Proof
Boy/Male
African, Arabic, Australian, British, English, German, Jamaican, Muslim, Pashtun, Sindhi, Swahili, Turkish
Kind; Merciful; Compassionate
Boy/Male
Hebrew
Ploughman. Son of Talmai (Talmai is a, meaning abounding in furrows.) Famous bearer: St...
Girl/Female
Hindu, Indian
Selected; Chosen
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR
MULTIVECTOR