When the quadratic form is definite, even though and may be determined from the above expressions, the set of vectors satisfying the defining equation may be empty, as is the case in a Euclidean space for a negative radius squared.
42.
You will need an expression for L ^ 2 * v ^ 2 in terms of L ^ 2-1 which can be easily obtained by inverting the defining equation for L . talk ) 16 : 48, 26 November 2012 ( UTC)
43.
The reason the circle can be represented by a graph in the neighborhood of every point is because the left hand side of its defining equation x ^ 2 + y ^ 2-1 = 0 has nonzero gradient at every point of the circle.
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In the context of a group action, this may be stated as : invariants are functions of coinvariants ( equivalence classes, orbits ), and a complete set of invariants characterizes the coinvariants ( is a set of defining equations for the coinvariants ).
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The Gibbs energy of formation of a compound is obtained from the defining equation ? " G " ?form = ? " H " ?form T ( ? " S " ?form ), and is expressed as
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Let " K " be a field over which the curve is defined ( i . e ., the coefficients of the defining equation or equations of the curve are in " K " ) and denote the curve by " E ".
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:I suppose the meaning intended here is that there is no definition in the form of an expression using only elementary functions, in other words, a closed-form solution of a defining equation for the Bernoulli numbers .-- Lambiam 06 : 38, 4 November 2007 ( UTC)
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As an application of this geometry, given an algebra that is a quotient of a polynomial algebra on k generators, it corresponds geometrically to an algebraic variety in k-dimensional space, and the weight must fall on the variety i . e ., it satisfies defining equations for the variety.
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Note that the defining equations for these statements are the same, as AJ = JA is the same as AJ-JA = 0, which is the same as [ A, J ] = 0, though the meaning of the Lie bracket vanishing is less immediate geometrically than the meaning of commuting.
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In light of this analogy, primary definitions are base quantities with no defining equation, but defined standardized condition, " secondary " definitions are quantities defined purely in terms of base quantities, " tertiary " for quantities in terms of both base and " secondary " quantities, " quaternary " for quantities in terms of base, " secondary ", and " tertiary " quantities, and so on.
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