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Vector part of a biquaternion, has three complex dimensions
a bivector is the vector part of a biquaternion. For biquaternion q = w + xi + yj + zk, w is called the biscalar and xi + yj + zk is its bivector part
Bivector_(complex)
Sum of directed areas in exterior algebra
quantity, a bivector is of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two
Bivector
Study of complex manifolds and several complex variables
of structures on complex spaces can be useful, in that it can allow one to solve classify the spaces themselves. Bivector (complex) Calabi–Yau manifold
Complex_geometry
Mathematical structure in differential geometry
algebras. Equivalently, recall that a holomorphic bivector field π {\displaystyle \pi } on a complex manifold M {\displaystyle M} is a section π ∈ Γ (
Poisson_manifold
Principal square root of minus 1
isomorphic to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric
Imaginary_unit
Matrix whose conjugate transpose is its negative (additive inverse)
{\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)} Bivector (complex) Hermitian matrix Normal matrix Skew-symmetric matrix Unitary matrix
Skew-Hermitian_matrix
Four-dimensional number system
to bivectors – quantities with magnitudes and orientations associated with particular 2D planes rather than 1D directions. The relation to complex numbers
Quaternion
Geometric space with six dimensions
tensor discussed in the previous section is a bivector in R 3 , 1 {\displaystyle \mathbb {R} ^{3,1}} . Bivectors can be used to generate rotations in either
Six-dimensional_space
Mathematical object that describes the electromagnetic field in spacetime
(sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a tensor that describes the electromagnetic field in spacetime. The
Electromagnetic_tensor
Geometric object used to describe rotation in any number of dimensions
geometric algebra, with the planes of rotations associated with simple bivectors in the algebra. Mathematically such planes can be described in a number
Plane_of_rotation
Element of an exterior algebra
following examples show that a bivector in two dimensions measures the area of a parallelogram, and the magnitude of a bivector in three dimensions also measures
Multivector
Set of lines described by homogeneous polynomial equations
p\wedge p=0} . Note that this is not a trivial condition, as a generic bivector does not satisfy it, for example ( e 1 ∧ e 2 + e 3 ∧ e 4 ) {\displaystyle
Line_complex
Setting of relativistic physics in geometric algebra
Spacetime algebra is a vector space that allows not only vectors, but also bivectors (directed quantities describing rotations associated with rotations or
Spacetime_algebra
Vector space with generalized dot product
scalar (a 0-vector), while the exterior product sends two vectors to a bivector (2-vector) – and in this context the exterior product is usually called
Inner_product_space
Every rigid motion is a screw displacement
rotation satisfying B 2 2 = − 1 {\displaystyle B_{2}^{2}=-1} . The two bivector lines B 1 {\displaystyle B_{1}} and B 2 {\displaystyle B_{2}} are orthogonal
Chasles'_theorem_(kinematics)
Algebraic structure designed for geometry
interpretation and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities of
Geometric_algebra
Concept in group theory (mathematics)
transformation can be decomposed into a commuting rotation and boost. Any bivector F {\displaystyle F} in the geometric algebra R p , q , r {\displaystyle
Invariant_decomposition
Quaternions with complex number coefficients
biscalar minus bivector is q ∗ = w − x i − y j − z k , {\displaystyle q^{*}=w-x\mathbf {i} -y\mathbf {j} -z\mathbf {k} \!\ ,} and the complex conjugation
Biquaternion
Part of a line that is bounded by two distinct end points; line with two endpoints
one-dimensional space, a ball is a line segment. An oriented plane segment or bivector generalizes the directed line segment. Beyond Euclidean geometry, geodesic
Line_segment
Textbook by E. B. Wilson based on the lectures of J. W. Gibbs
taught at Yale. First Wilson associates a bivector with an ellipse. The product of the bivector with a complex number on the unit circle is then called
Vector_Analysis
Algebra based on a vector space with a quadratic form
applications to differential geometry. Lounesto, Pertti (1993), "What is a bivector?", in Z. Oziewicz; B. Jancewicz; A. Borowiec (eds.), Spinors, Twistors
Clifford_algebra
Classification in abstract algebra
subalgebra. In both signatures the even subalgebra is generated by 1 and the bivector e1e2, and ( e 1 e 2 ) 2 = − e 1 2 e 2 2 = − 1. {\displaystyle
Classification of Clifford algebras
Classification_of_Clifford_algebras
Double cover Lie group of the special orthogonal group
spin algebra s p i n {\displaystyle {\mathfrak {spin}}} is defined as the bivector subalgebra Cl 2 = s p i n ( V ) = s p i n ( n ) , {\displaystyle \operatorname
Spin_group
Type of group in mathematics
characterization is used in interpreting the curl of a vector field (naturally a bivector) as an infinitesimal rotation or "curl", hence the name. The orthogonal
Orthogonal_group
Provides integral formulas for all derivatives of a holomorphic function
derivative a scalar part, the divergence ( k = 0 {\displaystyle k=0} ), and a bivector part, the curl ( k = 2 {\displaystyle k=2} ). This particular derivative
Cauchy's_integral_formula
this bivector is a well-defined scalar number representing the area of the parallelogram. (For vectors in three-dimensional space, the bivector-valued
Area_of_a_triangle
Non-tensorial representation of the spin group
scalar, 1, three orthogonal unit vectors, σ1, σ2 and σ3, the three unit bivectors σ1σ2, σ2σ3, σ3σ1 and the pseudoscalar i = σ1σ2σ3. It is straightforward
Spinor
Ways to represent 3D rotations
return. Bivectors in GA have some unusual properties compared to vectors. Under the geometric product, bivectors have a negative square: the bivector x̂ŷ
Rotation formulations in three dimensions
Rotation_formulations_in_three_dimensions
Concept in numerical linear algebra
child structures such as geometric algebras, rotations are represented by bivectors. Givens rotations are represented by the exterior product of the basis
Givens_rotation
Geometric space with four dimensions
_{24}+(a_{3}b_{4}-a_{4}b_{3})\mathbf {e} _{34}.\end{aligned}}} This is bivector valued, with bivectors in four dimensions forming a six-dimensional linear space with
Four-dimensional_space
Concept in differential geometry
} is a Poisson bivector on M {\displaystyle M} . Given a Courant algebroid with inner product of split signature, a generalized complex structure L → M
Courant_algebroid
Branch of mathematics
and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the shape operator. Below are some examples of how
Differential_geometry
Concept in general relativity
pp-wave if and only if it admits a covariantly constant bivector. (If so, this bivector is a null bivector.) It is a purely mathematical fact that the characteristic
Pp-wave_spacetime
Square root of the determinant of a skew-symmetric square matrix
}}}).} One can associate to any skew-symmetric 2n × 2n matrix A = (aij) a bivector ω = ∑ i < j a i j e i ∧ e j , {\displaystyle \omega =\sum _{i<j}a_{ij}\;e_{i}\wedge
Pfaffian
area A, S Extent of a two-dimensional geometric shape m2 L2 extensive, bivector or scalar area density ρA Mass per unit area kg⋅m−2 M L−2 intensive capacitance
List_of_physical_quantities
Type of manifold in differential geometry
makes any symplectic manifold into a Poisson manifold. The Poisson bivector is a bivector field π {\displaystyle \pi } defined by { f , g } = π ( d f ∧ d
Symplectic_manifold
American physicist and science educator
things, it reveals that the complex factor i ℏ {\displaystyle i\hbar } in the equation is a geometric quantity (a bivector) identified with electron spin
David_Hestenes
Relationship between relativity and pre-quantum electromagnetism
mathematical object with 6 components: an antisymmetric second-rank tensor, or a bivector. This is called the electromagnetic field tensor, usually written as Fμν
Classical electromagnetism and special relativity
Classical_electromagnetism_and_special_relativity
Example of a phase-space star product in mathematics
dimension 2n). To provide an explicit formula, consider a constant Poisson bivector Π on R 2 n {\displaystyle \mathbb {R} ^{2n}} : Π = ∑ i , j Π i j ∂ i ∧
Moyal_product
Force acting on charged particles in electric and magnetic fields
_{0}\right)\gamma _{0}} F {\displaystyle {\mathcal {F}}} is a spacetime bivector (an oriented plane segment, just like a vector is an oriented line segment)
Lorentz_force
Broad concept generalizing scalars in mathematics and physics
space, such as wind velocity over Earth's surface. Pseudo vectors and bivectors are also admitted as physical vector quantities. In mathematics, a vector
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Motion of a certain space that preserves at least one point
Minkowski quadratic form) the rotation of a vector space can be expressed as a bivector. This formalism is used in geometric algebra and, more generally, in the
Rotation_(mathematics)
Algebra associated to any vector space
that the exterior product is not an ordinary vector, but instead is a bivector. Bringing in a third vector w = w 1 e 1 + w 2 e 2 + w 3 e 3 , {\displaystyle
Exterior_algebra
Hypercomplex number system
σ 2 , σ 3 {\displaystyle \sigma _{1},\sigma _{2},\sigma _{3}} } are bivectors (e.g. γ { 1 , 2 , 3 } γ 0 {\displaystyle \gamma _{\{1,2,3\}}\gamma _{0}}
Octonion
Linear transformation of spacetime coordinates
or by using differential forms, which can be used to derive the Riemann bivector-valued 2-forms (aka tensor) and which can also treat moving frames. General
Biquaternion Lorentz transformation
Biquaternion_Lorentz_transformation
Formulations of electromagnetism
\nabla =\gamma ^{\mu }\partial _{\mu }.} The Riemann–Silberstein becomes a bivector F = E + I c B = E 1 γ 1 γ 0 + E 2 γ 2 γ 0 + E 3 γ 3 γ 0 − c ( B 1 γ 2 γ
Mathematical descriptions of the electromagnetic field
Mathematical_descriptions_of_the_electromagnetic_field
Eight-dimensional algebra over the real numbers
a quaternion as the sum of a scalar and a vector (strictly speaking a bivector), that is A = a0 + A, where a0 is a real number and A = A1 i + A2 j + A3
Dual_quaternion
Application of Clifford algebra
are axes for translations, and instead of having an algebra resembling complex numbers or quaternions, their algebraic behaviour is the same as the dual
Plane-based_geometric_algebra
Form of a matrix
V} with an inner product may be defined as the bivectors on the space, which are sums of simple bivectors (2-blades) v ∧ w . {\textstyle v\wedge w.} The
Skew-symmetric_matrix
Hamilton's original treatment of quaternions
called a biscalar. The vector part of a biquaternion is a bivector consisting of three complex components. The biquaternions are then the complexification
Classical Hamiltonian quaternions
Classical_Hamiltonian_quaternions
formalisms of special relativity. It uses mathematical objects such as bivectors to replace tensors in traditional formalisms of Minkowski spacetime, leading
Formulations of special relativity
Formulations_of_special_relativity
Vector satisfying some of the criteria of an eigenvector
necessary for the eigenvalues and the components of the eigenvectors to have complex values. The set spanned by all generalized eigenvectors for a given λ {\displaystyle
Generalized_eigenvector
Second order tensor in vector algebra
dyadic to related terms triadic, tetradic and polyadic. Kronecker product Bivector Polyadic algebra Unit vector Multivector Differential form Quaternions
Dyadics
Classification used in differential geometry and general relativity
as the Weyl tensor, evaluated at some event, as acting on the space of bivectors at that event like a linear operator acting on a vector space: X a b →
Petrov_classification
German-American mathematician and physicist (1908–1989)
(Pasadena, California Institute of Technology). 1944: With A. Einstein. "Bivector fields". Ann. Math. 45:1-14. 1945: "On the glancing reflection of shock
Valentine_Bargmann
Sum of a scalar and vector in Clifford algebra
1 {\displaystyle 1^{\dagger }=1} On the other hand, the trivector and bivectors change sign under reversion conjugation and are said to be purely imaginary
Paravector
Geometric object that has length and direction
dimensions, though the closely related exterior product does, whose result is a bivector. In two dimensions this is simply a pseudoscalar ( a 1 e 1 + a 2 e 2 )
Euclidean_vector
Type of motion
geometric algebra, with the planes of rotations associated with simple bivectors in the algebra. Mathematically such planes can be described in a number
Rotation_around_a_fixed_axis
Algebraic object with geometric applications
\mathbb {R} } . More generally, V can be taken over any field F (e.g. the complex numbers), with F replacing R {\displaystyle \mathbb {R} } as the codomain
Tensor
Complex vector of electromagnetic fields
and F was defined as a complexified 3-dimensional vector field, called a bivector field. The Riemann–Silberstein vector is used as a point of reference in
Riemann–Silberstein_vector
Conserved physical quantity; rotational analogue of linear momentum
also appears in the geometric algebra formalism, in which L and ω are bivectors, and the moment of inertia is a mapping between them. In relativistic
Angular_momentum
French mathematical physicist (1915–1998)
Lichnerowicz formulated the first definitions of a Poisson manifold in terms of a bivector, the counterpart of a (symplectic) differential 2-form. He showed later
André_Lichnerowicz
– vehō veh- vex- vect- carry advect, advection, advective, biconvex, bivector, circumvection, convect, convection, convective, convector, convex, convexity
List of Latin verbs with English derivatives
List_of_Latin_verbs_with_English_derivatives
Quantum mechanics taking into account particles near or at the speed of light
four-dimensional position and momentum of the particle, equivalently a bivector in the exterior algebra formalism: M α β = X α P β − X β P α = 2 X [ α
Relativistic quantum mechanics
Relativistic_quantum_mechanics
Unital algebra homomorphism of exterior algebras
{\displaystyle y} , z {\displaystyle z} , the outermorphism is linear over bivectors: f _ ( α x ∧ z + β y ∧ z ) = f _ ( ( α x + β y ) ∧ z ) = f ( α x + β y
Outermorphism
BIVECTOR COMPLEX
BIVECTOR COMPLEX
Boy/Male
Muslim
The guide, Director, Leader
Male
Egyptian
, a royal scribe and director of soldiers.
Boy/Male
Muslim/Islamic
Director guide
Boy/Male
Celebrity, Hindu, Indian, Tamil
Lord Ayyappa's Related Name; Famous Tamil Actor and Director Name also
Surname or Lastname
English
English : nickname for someone with a complexion that was as ‘white as a lily’ (Middle English lilie).
Male
Egyptian
, chief, director.
Boy/Male
Muslim
The guide, Director, Leader
Boy/Male
Arabic, Muslim
Guiding to the Right; Director; Guide
Surname or Lastname
English
English : from the popular medieval personal name Hudde, which is of complex origin. It is usually explained as a pet form of Hugh, but there was a pre-existing Old English personal name, Hūda, underlying place names such as Huddington, Worcestershire. This personal name may well still have been in use at the time of the Norman Conquest. If so, it was absorbed by the Norman Hugh and its many diminutives. Reaney adduces evidence that Hudde was also regarded as a pet form of Richard.German : from a short form of a Germanic compound personal name formed with hut ‘guard’ as the first element.Variant spelling of German Hütt (see Huett).Jewish (Ashkenazic) : metonymic occupational name from Yiddish hut, German Hut ‘hat’ (see Huth).
Boy/Male
Indian
The guide, Director, Leader
Girl/Female
Arabic, Muslim
A Director; A Leader; A Guide
Boy/Male
Arabic, French
Director; Leader; Guide to Righteousness
Surname or Lastname
English
English : nickname for a person with a ruddy complexion, from an adjective derivative of Middle English mad(d)er ‘madder’, the dye plant (see Mader 1), here used in a transferred sense.
Boy/Male
Indian
The guide, Director, Leader
Boy/Male
Indian
The guide, Director, Leader
Boy/Male
Muslim
The guide, Director, Leader
Male
English
This name became popular as a boy's name after the making of the Star Wars saga by George Lucas, who named his Darth Vader character after the surname of director Ken Annakin, a variant spelling of the Low German female personal name Anniken, a form of Hannah, ANAKIN means "favor; grace."
Boy/Male
Arabic, Australian, Muslim
Director; Leader; Guide to Righteousness
Boy/Male
Arabic, Muslim
Princes; Lords; Chiefs Title of the Prophet Muhammad's Director Descendants
Surname or Lastname
German
German : nickname from the small medieval coin known as the häller or heller because it was first minted (in 1208) at the Swabian town of (Schwäbisch) Hall. Compare Hall.Jewish (Ashkenazic) : habitational name for someone from Schwäbisch Hall.German : topographic name for someone living by a field named as ‘hell’ (see Helle 3).English : topographic name for someone living on a hill, from southeastern Middle English hell + the habitational suffix -er.Dutch : from a Germanic personal name composed of the elements hild ‘strife’ + hari, heri ‘army’.Jewish (Ashkenazic) : nickname for a person with fair hair or a light complexion, from an inflected form, used before a male personal name, of German hell ‘light’, ‘bright’, Yiddish hel.
BIVECTOR COMPLEX
BIVECTOR COMPLEX
Surname or Lastname
English
English : variant of Mixon 2.
Boy/Male
Indian, Punjabi, Sikh
Lamp of Air
Girl/Female
French
Forerunner of Alice. Of the nobility. Noble.
Girl/Female
Indian
Reflection, Image, Radiance
Girl/Female
Indian, Punjabi, Sikh
A Brave Godly Person
Boy/Male
Hindu
Girl/Female
Muslim/Islamic
Friendly
Boy/Male
Muslim/Islamic
It was the name of the Tabiee Abu Salih
Surname or Lastname
English
English : patronymic from Good.
Boy/Male
Indian, Punjabi, Sikh
Protector of the Morning
BIVECTOR COMPLEX
BIVECTOR COMPLEX
BIVECTOR COMPLEX
BIVECTOR COMPLEX
BIVECTOR COMPLEX
n.
Same as Radius vector.
n.
One who sings the leading part; the director or leader.
n.
A director; one who gives a mandate or order.
n.
One who ordains or establishes; a director.
n.
One who, or that which, directs; one who regulates, guides, or orders; a manager or superintendent.
n.
A part of a machine or instrument which directs its motion or action.
n.
A director.
n.
A term made up of the two parts / + /1 /-1, where / and /1 are vectors.
n.
The condition or office of a director; directorate.
n.
One who directs; a director.
v. t.
A grooved director for a probe or knife.
a.
Having the quality of a director, or authoritative guide; directive.
n.
The leader or director of an orchestra or chorus.
n.
One of a body of persons appointed to manage the affairs of a company or corporation; as, the directors of a bank, insurance company, or railroad company.
n.
A guide or director.
n.
A guide; a director.
n.
One who, or that which, bisects; esp. (Geom.) a straight line which bisects an angle.
n.
A spiritual overseer, superintendent, or director.
n.
A slender grooved instrument upon which a knife is made to slide when it is wished to limit the extent of motion of the latter, or prevent its injuring the parts beneath.
n.
A directed quantity, as a straight line, a force, or a velocity. Vectors are said to be equal when their directions are the same their magnitudes equal. Cf. Scalar.