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The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.
An open formula is a formula that contains at least one free variable. [citation needed] An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like true or false. An open formula can be transformed into a closed formula by applying a ...
A key consequence of this is that "the integral of a closed form over homologous chains is equal": If ω is a closed k-form and M and N are k-chains that are homologous (such that M − N is the boundary of a (k + 1)-chain W), then =, since the difference is the integral = =.
One example is the planetary movement of three bodies: while there is no closed-form solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos. Formally, a Hamiltonian system is a dynamical system characterised by the scalar function H ( q , p , t ) {\displaystyle H({\boldsymbol {q}},{\boldsymbol ...
In physics and chemistry, an equation of state is a thermodynamic equation ... Closed system; Open system; ... The general form of an equation of state may be ...
In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball in R n is exact for p with 1 ≤ p ≤ n. [1] The lemma was introduced by Henri Poincaré in 1886. [2] [3]
In the differential form formulation on arbitrary space times, F = 1 / 2 F αβ dx α ∧ dx β is the electromagnetic tensor considered as a 2-form, A = A α dx α is the potential 1-form, = is the current 3-form, d is the exterior derivative, and is the Hodge star on forms defined (up to its orientation, i.e. its sign) by the ...