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Ternary operation on vectors
scalar triple product and, less often, the vector-valued vector triple product. The scalar triple product (also called the mixed product, box product, or
Triple_product
Topics referred to by the same term
Triple product is a ternary operation on vectors. It may also mean: Jacobi triple product, an identity in number theory Triple product rule, a calculus
Triple product (disambiguation)
Triple_product_(disambiguation)
Criterion for igniting a nuclear fusion chain reaction
useful figure of merit is the triple product of density, confinement time, and plasma temperature T. The triple product also has a minimum required value
Lawson_criterion
Mathematical identity found by Jacobi in 1829
In mathematics, the Jacobi triple product is the identity: ∏ m = 1 ∞ ( 1 − x 2 m ) ( 1 + x 2 m − 1 y 2 ) ( 1 + x 2 m − 1 y 2 ) = ∑ n = − ∞ ∞ x n 2 y 2
Jacobi_triple_product
Relation between relative derivatives of three variables
The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule, Euler's chain rule, or the reciprocity theorem, is a
Triple_product_rule
Topics referred to by the same term
Dot product Cross product Seven-dimensional cross product Triple product, in vector calculus Tensor product Product topology Cap product Cup product Slant
Product
Mathematical operation on vectors in 3D space
} The vector triple product is the cross product of a vector with the result of another cross product, and is related to the dot product by the following
Cross_product
Algebraic operation on coordinate vectors
There are two ternary operations involving dot product and cross product. The scalar triple product of three vectors is defined as a ⋅ ( b × c ) = b
Dot_product
In abstract algebra, the triple product property is an identity satisfied in some groups. Let G {\displaystyle G} be a non-trivial group. Three nonempty
Triple_product_property
application of the product-to-sum formulae, and beat (acoustics) and phase detector for applications of the sum-to-product formulae. The product of two sines
List of trigonometric identities
List_of_trigonometric_identities
Algebra associated to any vector space
^{3})} . The scalar coefficient is the triple product of the three vectors. The cross product and triple product in three dimensions each admit both geometric
Exterior_algebra
Geometric object that has length and direction
cross product of two vectors is a pseudovector instead of a vector (see below). The scalar triple product (also called the box product or mixed triple product)
Euclidean_vector
Lie triple systems and Jordan triple systems. They were introduced by Nathan Jacobson in 1949. In particular, any Lie algebra defines a Lie triple system
Triple_system
Operation in algebraic topology
)+H^{*}(\Gamma )[w]).} So the Massey product can be regarded as a function defined on triples of classes such that the product of the first or last two is zero
Massey_product
Topics referred to by the same term
Box product may refer to: The scalar triple product of three vectors A cartesian product of topological spaces equipped with the box topology The cartesian
Box_product
Measure in 3-dimensional geometry
denotes the scalar triple product of the three vectors and a → ⋅ b → {\displaystyle {\vec {a}}\cdot {\vec {b}}} denotes the scalar product. Care must be taken
Solid_angle
Variety of Curaçao liqueur
traditional sugar beet or cane sugar, the product is sweetened with agave syrup.[better source needed] Triple sec is usually made from a spirit derived
Triple_sec
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
Δ r i ) ) ] … vector triple product = − [ ( ω × Δ r i ) ( ω ⋅ Δ r i ) − Δ r i ( Δ r i ⋅ ( ω × ω ) ) ] … scalar triple product = − [ ( ω × Δ r i ) (
Moment_of_inertia
Reaction that combines atomic nuclei
typically occurs via thermonuclear fusion, an isotropic process requiring a triple product of very high temperature (in the kiloelectronvolt or hundred million
Nuclear_fusion
Japanese tokamak
of the fusion triple product achieved: 1.77×1028 K·s·m−3 = 1.53×1021 keV·s·m−3. The product quoted is not a valid fusion triple product since the plasmas
JT-60
Mathematical function
while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan
Ramanujan_theta_function
Scalar quantity, changing sign in mirrored coordinates
which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where
Pseudoscalar
American fusion research company
Experiments that year achieved plasmas with multi-keV temperatures and a triple product of 6.4 × 1018 keV·s/m3. In 2021, the firm announced that after a 16-month
Helion_Energy
Observable quantity in imaging astronomical interferometry
"closure phase" observations is actually the complex quantity called the triple product (or bispectrum). The closure phase is the phase (argument) of this complex
Closure_phase
Formulas about vectors in three-dimensional Euclidean space
three-dimensional vectors, this is equal to the square of the scalar triple product det [ A , B , C ] = | A , B , C | {\displaystyle \det[\mathbf {A} ,\mathbf
Vector_algebra_relations
Calculus of vector-valued functions
basic algebraic operations consist of: Also commonly used are the two triple products: Vector calculus studies various differential operators defined on
Vector_calculus
Way to determine a preliminary orbit from initial observations in astronomy
scalar quantity (scalar triple product), take the dot product of the first observational unit vector with the cross product of the second and third observational
Gauss's_method
Infinite product identity introduced by Watson
rediscovered by Bailey (1951) and Gordon (1961). It is analogous to the Jacobi triple product identity, and is the Macdonald identity for a certain non-reduced affine
Quintuple_product_identity
the product of two cubes, {4,3} × {4,3}. In geometry of 6 dimensions or higher, a triple product is a polytope resulting from the Cartesian product of
Proprism
, then ( x , y , z ) {\displaystyle (x,y,z)} may denote the scalar triple product.[citation needed] See also [□,□,□] in § Square brackets. (□, ..., □)
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Media that are encoded in machine-readable formats
and culture. Digital media platforms like YouTube operate through a triple-product business model in which they provide information and entertainment (infotainment)
Digital_media
Former experimental tokamak at Princeton Plasma Physics Laboratory
set several records for power output, maximum temperature, and fusion triple product. TFTR shut down in 1997 after fifteen years of operation. PPPL used
Tokamak_Fusion_Test_Reactor
How many ways a positive integer can be represented as the sum of four squares
theorem can be proved by elementary means starting with the Jacobi triple product. The proof shows that the Theta series for the lattice Z4 is a modular
Jacobi's_four-square_theorem
Index of articles associated with the same name
{\displaystyle (m\times n)} matrix. Triple products – products involving three vectors. Quadruple products – products involving four vectors. Vector multiplication
Vector_multiplication
Fusion power device
together. The product of temperature, pressure and time produces the expected rate of fusion events, the so-called fusion triple product. To be useful
Spherical_tokamak
Mathematical operation
square root. An example of a ternary operation (arity three) is the triple product. The term algebraic operation may also be used for operations that may
Algebraic_operation
Topics referred to by the same term
Reciprocity law for Dedekind sums Betti's theorem in linear elasticity The Triple Product rule, a formula relating the partial derivatives of three independent
Reciprocity_theorem
Geometric property of objects being in the same plane
plane. This leads to the following coplanarity test using a scalar triple product: Four distinct points, x1, x2, x3, x4, are coplanar if and only if,
Coplanarity
Infinite product identities associated to affine root systems
infinite product identities associated to affine root systems, introduced by Ian Macdonald (1972). They include as special cases the Jacobi triple product identity
Macdonald_identities
Mathematical concept
mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product ∏ n = 1 ∞ a n = a 1 a 2 a 3 ⋯ {\displaystyle \prod _{n=1}^{\infty
Infinite_product
Geometry of figures on the surface of a sphere
z-axis along OB→, the triple product OB→ · (OC→ × OA→), evaluates to sin c sin a sin B. Therefore, the invariance of the triple product under cyclic permutations
Spherical_trigonometry
Type of infinitesimal in calculus
{\displaystyle z} . The cyclic relation is also known as the cyclic rule or the Triple product rule. Setting the second term in brackets equal to zero yields ( ∂ z
Exact_differential
Property of all triangles on a Euclidean plane
={\begin{pmatrix}\sin b\cos A\\\sin b\sin A\\\cos b\end{pmatrix}}.} The scalar triple product, OA ⋅ (OB × OC) is the volume of the parallelepiped formed by the position
Law_of_sines
Description of how businesses operate
Society. doi:10.1177/14614448231207644. ISSN 1461-4448. "Figure 3. The triple-product business model of digital media markets." Borgh, Michel; Cloodt, Myriam;
Business_model
Fusion power research concept
Together, this is termed the triple product (see Lawson criterion). Fusion research focuses on reaching the highest triple product possible. Magnetic fusion
Magnetized_target_fusion
Theorem in vector calculus
^{3}} We can now recognize the difference of partials as a (scalar) triple product: ∂ P v ∂ u − ∂ P u ∂ v = ∂ ψ ∂ v ⋅ ( ∇ × F ) × ∂ ψ ∂ u = ( ∇ × F ) ⋅
Stokes'_theorem
Topics referred to by the same term
other formulas/equations: Cyclic chain rule (AKA Euler's chain rule, triple product rule, cyclic relation, reciprocity theorem), relating partial derivatives
Chain_rule_(disambiguation)
Topics referred to by the same term
Louis Lagrange: Lagrange interpolation formula Lagrange–Bürmann formula Triple product expansion Mean value theorem Euler–Lagrange equation This disambiguation
Lagrange's_formula
Angle between two planes in space
previous formulas by using the vector quadruple product formula, and the fact that a scalar triple product is zero if it contains twice the same vector:
Dihedral_angle
Jacques (2001). "p-adic measures and square roots of special values of triple product L-functions". Mathematische Annalen. 320 (1): 127–147. arXiv:math/9610228
Jacques_Tilouine
Characteristic quantity of plasmas
are combined in the Lawson criterion, or its modern form, the fusion triple product. In order to be efficient, the rate of fusion energy being deposited
Plasma_beta
Force acting on charged particles in electric and magnetic fields
because a vector is always perpendicular to its cross product with another vector; the scalar triple product v ⋅ ( v × B ) {\displaystyle \mathbf {v} \cdot (\mathbf
Lorentz_force
1829 book on mathematics by Carl G.J. Jacobi
occasions. The book introduces Jacobi elliptic functions and the Jacobi triple product identity. One of the most exciting moments of my life was when, after
Fundamenta nova theoriae functionum ellipticarum
Fundamenta_nova_theoriae_functionum_ellipticarum
Math operation methods
× 3 inverse can be expressed concisely in terms of the cross product and triple product. If a matrix A = [ x 0 x 1 x 2 ] {\displaystyle \mathbf {A}
Methods_of_matrix_inversion
Graphical means of performing computations in linear algebra
the cross product and the dot product, one can read off the three leftmost diagrams as precisely the three leftmost scalar triple products in the above
Trace_diagram
Modern stellarator for plasma fusion experiments
particles/m3, and a confinement time of 0.2 second yielded a record fusion product of 6 × 1026 degree-seconds per cubic metre. During the last experiments
Wendelstein_7-X
Approach to controlled thermonuclear fusion using magnetic fields
region for a sufficient time, as specified by the Lawson criterion (triple product). The high temperature of a fusion plasma precludes the use of material
Magnetic_confinement_fusion
Integer side lengths of a right triangle
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known
Pythagorean_triple
Turning force around an axis
}}\times \mathbf {r} } The expression inside the integral is a scalar triple product F ⋅ d θ × r = r × F ⋅ d θ {\displaystyle \mathbf {F} \cdot \mathrm {d}
Torque
American security contractor
Triple Canopy, Inc., was an American private security company and private military company (PMC) that provides armed security, mission support, and risk
Triple_Canopy
Characterization by prime factors of sums of two squares
Gaussian integers. Hirschhorn gives a short proof derived from the Jacobi triple product. Legendre's three-square theorem Lagrange's four-square theorem Sum
Sum_of_two_squares_theorem
Mathematical concept
and are not satisfied by the seven-dimensional cross product, notably, Vector triple product: x × ( y × z ) = ( x ⋅ z ) y − ( x ⋅ y ) z {\displaystyle
Seven-dimensional cross product
Seven-dimensional_cross_product
Flight or sailing route along the shortest path between two points on a globe's surface
the convoluted expression of s⊥, the evaluation may employ that the triple product is invariant under a circular shift of the three arguments: cos p
Great-circle_navigation
Formula in calculus
formula Product rule – Formula for the derivative of a product Quotient rule – Formula for the derivative of a ratio of functions Triple product rule –
Chain_rule
Differential operator in mathematics
can be seen to be a special case of Lagrange's formula; see Vector triple product. For expressions of the vector Laplacian in other coordinate systems
Laplace_operator
Number of arguments required by a function
considered as a function of a single argument that is an element of some product space. However, it may be convenient for notation to consider n-ary functions
Arity
Media consisting of both information and entertainment
Society. doi:10.1177/14614448231207644. ISSN 1461-4448. "Figure 3. The triple-product business model of digital media markets." Wikimedia Commons has media
Infotainment
Polyhedron with four faces
given vertices, another approach is by the absolute value of the scalar triple product, representing the absolute values of determinants 6 ⋅ V = | a b c |
Tetrahedron
Power released by a short pulse
inertial fusion) program on Z has achieved the second-highest fusion triple product recorded. Several private companies are also developing pulsed power
Pulsed_power
Geometry and crystallography point array
b. The volume of the unit cell can be calculated by evaluating the triple product a · (b × c), where a, b, and c are the lattice vectors. The properties
Bravais_lattice
Antibiotic medication
acetate. The generic name for these products, regardless of the base, is "triple antibiotic ointment". In China, this product (with lidocaine HCl) is named
Neomycin/polymyxin B/bacitracin
Neomycin/polymyxin_B/bacitracin
Coefficients coupled with angular momentum
tensor direct product can be reduced to a sum of irreps, giving rise to the Clebsch-Gordon coefficients, or by reduction of the triple product of three irreps
3-j_symbol
Hexahedron with parallelogram faces
{b} \right)\cdot \mathbf {c} \right|.} The mixed product of three vectors is called triple product. It can be described by a determinant. Hence for a
Parallelepiped
point and line are just considered as ordered triples, their incidence is expressed as having their dot product equal 0. Let P1 and P2 be a pair of distinct
Incidence_(geometry)
Textbook by E. B. Wilson based on the lectures of J. W. Gibbs
products for pairs of vectors. These are extended to a scalar triple product and a quadruple product. Pages 77–81 cover the essentials of spherical trigonometry
Vector_Analysis
Topics referred to by the same term
paths between each pair of a, b, and c Median algebra, an algebraic triple product generalising the algebraic properties of the majority function Median
Median_(disambiguation)
Probability distribution on the circle
wrapped normal distribution may also be expressed in terms of the Jacobi triple product: f WN ( θ ; μ , σ ) = 1 2 π ∏ n = 1 ∞ ( 1 − q n ) ( 1 + q n − 1 / 2
Wrapped_normal_distribution
Concept in differential geometry
the products of the triple product of the spin and U(1) component bundles, are either 12 = 1 or (−1)2 = 1 and so the spinC bundle satisfies the triple overlap
Spin_structure
Vector formula for a rotation in space, given its axis
\times \mathbf {v} )} , where the last equality follows from the vector triple product formula: a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c {\textstyle \mathbf
Rodrigues'_rotation_formula
Mathematical function
so that the series expansion has integral coefficients. The Jacobi triple product implies that the eta is (up to a factor) a Jacobi theta function for
Dedekind_eta_function
German mathematician (1804–1851)
of theta functions, including the functional equation and the Jacobi triple product formula, as well as many other results on q-series and hypergeometric
Carl_Gustav_Jacob_Jacobi
Shape formed from points common to other shapes
using the rules of a scalar triple product. If the scalar triple product equals to 0, then planes either do not have the triple intersection or it is a line
Intersection_(geometry)
Derivative of a function with multiple variables
differentiation Partial differential Symmetry of second derivatives Triple product rule, also known as the cyclic chain rule. Cajori, Florian (1952), A
Partial_derivative
Topics referred to by the same term
Jacobi's formula for the derivative of the determinant of a matrix Jacobi triple product, an identity in the theory of theta functions Jacobi's theorem (disambiguation)
Jacobi
Homotopy operation
Whitehead product operation are called a graded quasi-Lie algebra; this is proven in Uehara & Massey (1957) via the Massey triple product. If f ∈ π 1
Whitehead_product
How many integer lattice points there are in a circle
two-square theorem, which follows almost immediately from the Jacobi triple product. A much simpler sum appears if the sum of squares function r 2 ( n )
Gauss_circle_problem
Mathematical strategy
{v}}\times {\vec {q}})\\\end{aligned}}} and upon substitution for the triple product q v q ∗ = ( q w , q → ) ( v → ⋅ q → , q w v → − v → × q → ) = ( 0 ,
Conversion between quaternions and Euler angles
Conversion_between_quaternions_and_Euler_angles
Theorem in calculus
\mathbf {c} )\cdot \mathbf {n} \,\mathrm {d} S.} By reordering the triple product on the right hand side and taking out the constant vector of the integral
Divergence_theorem
function Jacobi's theorem (skew-symmetric matrix) Jacobi transform Jacobi triple product Jacobi-type J-fractions Last geometric statement of Jacobi Generalized
List of things named after Carl Gustav Jacob Jacobi
List_of_things_named_after_Carl_Gustav_Jacob_Jacobi
High-budget video game
In the video game industry, AAA, also known as triple-A, is a term used to classify video games produced or distributed by a mid-sized or major publisher
AAA_(video_game_industry)
to profit, whereas triple top line is a focus to align sustainability and business profitability from the inception of a product. The TTL approach is
Triple_top_line
Special functions of several complex variables
be everywhere defined, such as fields of p-adic numbers. The Jacobi triple product (a special case of the Macdonald identities) tells us that for complex
Theta_function
Chinese mathematician (born 1981)
Mathematics 186 (2017), No. 3, 767–911. (with Xinyi Yuan, Shou-Wu Zhang) "Triple product L-series and Gross–Kudla–Schoen cycles" Archived 11 April 2016 at the
Wei_Zhang_(mathematician)
American figure skater (born 1999)
U.S. Junior champion. She is the fourth American woman to land a clean triple axel in international competition. Glenn is the first American woman to
Amber_Glenn
Physical law
be rewritten in the following equivalent way by expanding the vector triple product and applying Stokes' theorem: F 12 = − μ 0 4 π ∫ L 1 ∫ L 2 ( I 1 d ℓ
Ampère's_force_law
Brand of triple sec
to refer to the triple sec liqueur itself, the specific term Cointreau L'Unique may be used to distinguish it from related products, such as the aforementioned
Cointreau
Form of fusion power
the surroundings. This concept is best expressed as the fusion triple product, the product of the temperature, density and "confinement time", the amount
Aneutronic_fusion
Proposed fusion reactor
confinement time of 25 seconds in 1982, as well as the record fusion triple product (density × energy-confinement-time × mean energy) of 4 × 1014 keV sec cm−3
Migma
) = x ( y ( x x ) ) {\displaystyle (xy)(xx)=x(y(xx))} . The Jordan triple product is defined by { a , b , c } = ( a b ) c + ( c b ) a − ( a c ) b . {\displaystyle
Mutation_(algebra)
Mechanism by which a celestial body generates a magnetic field
equivalent up to a boundary term (and with the double use of the scalar triple product identity) to − u ⋅ ( 1 μ 0 ( ∇ × B ) × B ) = − u ⋅ ( J × B ) {\textstyle
Dynamo_theory
travel, tourism, insurance
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