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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 (+,,, /).
The form generates the de Rham cohomology group ({}), meaning that any closed form is the sum of an exact form and a multiple of : = + , where = accounts for a non-trivial contour integral around the origin, which is the only obstruction to a closed form on the punctured plane (locally the derivative of a potential function) being the ...
Closed-form expression, a finitary expression Closed differential form , a differential form α {\displaystyle \alpha } whose exterior derivative d α {\displaystyle d\alpha } is the zero form 0 {\displaystyle 0} , meaning d α = 0 {\displaystyle d\alpha =0} .
Conversely, if closed sets are given and every intersection of closed sets is closed, then one can define a closure operator C such that () is the intersection of the closed sets containing X. This equivalence remains true for partially ordered sets with the greatest-lower-bound property , if one replace "closed sets" by "closed elements" and ...
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 = =.
Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions , exponential generating functions , Lambert series , Bell series , and Dirichlet series .
The field F is algebraically closed if and only if every rational function in one variable x, with coefficients in F, can be written as the sum of a polynomial function with rational functions of the form a/(x − b) n, where n is a natural number, and a and b are elements of F.
Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...