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This definition relies on the fact that every simple closed curve admits a well-defined interior, which follows from the Jordan curve theorem. The inner loop of a beltway road in a country where people drive on the right side of the road is an example of a negatively oriented ( clockwise ) curve.
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space , right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also ...
Since in Green's theorem = (,) is a vector pointing tangential along the curve, and the curve C is the positively oriented (i.e. anticlockwise) curve along the boundary, an outward normal would be a vector which points 90° to the right of this; one choice would be (,).
A torus is an orientable surface The Möbius strip is a non-orientable surface. Note how the disk flips with every loop. The Roman surface is non-orientable.. In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". [1]
A curve in the Euclidean plane is defined to be an Osgood curve when it is non-self-intersecting (that is, it is either a Jordan curve or a Jordan arc) and it has positive area. [1] More formally, it must have positive two-dimensional Lebesgue measure. Osgood curves have Hausdorff dimension two, like space-filling curves.
An example of a 1-dimensional manifold is an interval [a, b], and intervals can be given an orientation: they are positively oriented if a < b, and negatively oriented otherwise. If a < b then the integral of the differential 1 -form f ( x ) dx over the interval [ a , b ] (with its natural positive orientation) is
The convention for positive linking number is based on a right-hand rule. The winding number of an oriented curve in the x-y plane is equal to its linking number with the z-axis (thinking of the z-axis as a closed curve in the 3-sphere). More generally, if either of the curves is simple, then the first homology group of its complement is ...
A trihedron is said to be adapted to a surface if P always lies on the surface and e 3 is the oriented unit normal to the surface at P. In the case of the Darboux frame along an embedded curve, the quadruple (P(s) = γ(s), e 1 (s) = T(s), e 2 (s) = t(s), e 3 (s) = u(s)) defines a tetrahedron adapted to the surface into which the curve is embedded.