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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.
A Riemann surface is planar if and only if every closed 1-form of compact support is exact. [3] Let ω be a closed 1-form of compact support on a planar Riemann surface. If γ is a closed Jordan curve on the surface, then it separates the surface. Hence ∫ γ ω = 0. Since this is true for all closed Jordan curves, ω must be exact.
A plane simple closed curve is also called a Jordan curve. It is also defined as a non-self-intersecting continuous loop in the plane. [9] The Jordan curve theorem states that the set complement in a plane of a Jordan curve consists of two connected components (that is the curve divides the plane in two non-intersecting regions that are both ...
By the Jordan curve theorem, a simple closed curve divides the plane into interior and exterior regions, and another equivalent definition of a closed convex curve is that it is a simple closed curve whose union with its interior is a convex set. [9] [17] Examples of open and unbounded convex curves include the graphs of convex functions.
This is a list of Wikipedia articles about curves used in different fields: mathematics (including geometry, statistics, and applied mathematics), ...
If γ is a continuous closed curve on a Riemann surface X, there is a smooth closed 1-form α of compact support such that ∫ γ ω = ∫ X ω ∧ α for any closed smooth 1-form ω on X. The form α is unique up to adding an exact form and can be taken to have support in any open neighbourhood of the image of γ.
A general 1-form is a linear combination of these differentials at every point on the manifold: + +, where the f k = f k (x 1, ... , x n) are functions of all the coordinates. A differential 1-form is integrated along an oriented curve as a line integral.