Cartesian product en · NOUN
Etymology
From Cartesian + product, after French philosopher, mathematician, and scientist René Descartes (1596–1650), whose formulation of analytic geometry gave rise to the concept.
Meanings
- (of two sets X and Y) The set of all possible ordered pairs of elements, the being first from X, the second from Y, written X×Y. Formally, the set (x,y);|;x∈X;and;y∈Y.
- All possible combinations of rows between all of the tables listed.
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(of an m-dimensional space X and an n-dimensional space Y) An (m+n)-dimensional space, formally composed of all possible ordered pairs of points from X and Y, but thought of as an independent (m+n)-dimensional space (in the sense that if, e.g. X and Y are vector spaces, the elements of X×Y are thought of as (m+n)-tuples instead of ordered pairs) and written X×Y.
On the Cartesian product of two manifolds a differentiable structure can be constructed in the following way.
1987, M. Göckeler, T. Schücker, Differential Geometry, Gauge Theories, and Gravity, published 1989, page 98:
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Any of several generalizations of the set-theoretic sense, especially one which shares the geometrical intuition outlined above, i.e. one such that the product can be thought of as an object in its own right and not just as a set of pairs.
The hypercube is the simplest example of a Cartesian product of graphs; indeed, the m-hypercube is nothing but (K₂)ᵐ.
1997, Michel Marie Deza, Monique Laurent, Geometry of Cuts and Metrics, published 2009, page 297:A moment's thought convinces us of the following: The Cartesian product of two Riemannian Einstein metrics with the same constant Ricci scalar ρ is again Ricci-constant, and has Ric
2004, David Bao, Colleen Robles, “Ricci and Flag Curvatures in Finsler Geometry”, in David Dai-Wai Bao, Robert L. Bryant, Shiing-Shen Chern, Zhomgmin Shen, editors, A Sampler of Riemann-Finsler Geometry, page 246:
Forms
| Spelling | Features | Labels | Source |
|---|---|---|---|
| Cartesian products | Number=Plur | lexicographic | |
| cartesian product | alternative | lexicographic |
Synonyms
direct product (set of possible pairs)
Translations (26)
pl produkt kartezjański (set of possible pairs) · ja 直積集合 (set of possible pairs) · pl iloczyn kartezjański (set of possible pairs) · de kartesisches Produkt (set of possible pairs) · pt produto cartesiano (set of possible pairs) · fi karteesinen tulo (set of possible pairs) · es producto cartesiano (set of possible pairs) · cs kartézský součin (set of possible pairs) · ru прямо́е произведе́ние (set of possible pairs) · nl cartesisch product (set of possible pairs) · et otsekorrutis (set of possible pairs) · cmn 笛卡兒積 /笛卡儿积 (set of possible pairs) · ja デカルト積 (set of possible pairs) · sh Kartezijev produkt (set of possible pairs) · et Cartesiuse korrutis (set of possible pairs) · el καρτεσιανό γινόμενο (set of possible pairs) · fi tulojoukko (set of possible pairs) · et ristikorrutis (set of possible pairs) · mi whātuinga takirua (set of possible pairs) · nl kruisproduct (set of possible pairs) · de Kreuzprodukt (set of possible pairs) · it prodotto cartesiano (set of possible pairs) · cmn 笛卡爾乘積 /笛卡尔乘积 (set of possible pairs) · ro produs cartezian (set of possible pairs) · fr produit cartésien (set of possible pairs) · ru дека́ртово произведе́ние (set of possible pairs)