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As the exterior product is associative brackets are not needed as it does not matter which of a ∧ b or b ∧ c is calculated first, though the order of the vectors in the product does matter. Geometrically the trivector a ∧ b ∧ c corresponds to the parallelepiped spanned by a , b , and c , with bivectors a ∧ b , b ∧ c and a ∧ c ...
Pages with many brackets or large brackets, especially those containing flag icons, may come close to or exceed Wikipedia's Post-expand include size limit. In these cases consider using modules directly instead: {{32TeamBracket}} can be replaced with {{#invoke:bracket|32TeamBracket}}.
In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by the equivalent sum of products, continuing until the expression becomes a ...
The trinomial expansion can be calculated by applying the binomial expansion twice, setting = +, which leads to (+ +) = (+) = = = = (+) = = = ().Above, the resulting (+) in the second line is evaluated by the second application of the binomial expansion, introducing another summation over the index .
Two Hilbert spaces V and W may form a third space V ⊗ W by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.
This template also allows the tagger to add a brief explanation of what needs expansion in that section, something that {} did not. For a myriad of reasons, including the points stated above, the "Expand" template was finally deleted in January 2011 (three weeks after this was written) after a very long Templates for Discussion and an even ...
The brackets are much larger than those in North American professional leagues—while no more than 16 teams qualify for the postseason in any major North American league (this is the case in the NBA and NHL), 68 teams (out of over 350) advance to the NCAA men's tournament, with most bracket contests involving 64 of these teams.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.