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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 space curve is a curve for which is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves , although the above definition of a curve does not apply (a real algebraic curve may be disconnected ).
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 geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform , the name given to these shapes by Leonhard Euler . [ 1 ]
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} .
When I is a closed interval [a,b], γ(a) is called the starting point and γ(b) is the endpoint of γ. If the starting and the end points coincide (that is, γ(a) = γ(b)), then γ is a closed curve or a loop. To be a C r-loop, the function γ must be r-times continuously differentiable and satisfy γ (k) (a) = γ (k) (b) for 0 ≤ k ≤ r.
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.
Hyperbolic curves, also called curves of general type. See algebraic curves for examples. The classification of smooth curves can be refined by the degree for projectively embedded curves, in particular when restricted to plane curves. Note that all complete smooth curves are projective in the sense that they admit embeddings into projective ...