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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]
Given a surface S with a specified normal direction n̂ (a choice of "upward direction" with respect to S), the boundary curve C around S is defined to be positively oriented provided that the right thumb points in the direction of n̂ and the fingers curl along the orientation of the bounding curve C.
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 ...
e 2 is the result of rotating e 1 through an angle of 90° (where the sense is measured by assuming the pair e 1, e 2 to be positively oriented), and then rescaling so that e 2 ⋅ e 2 = 1 holds. e 1 is the result of rotating e 2 through an angle of 90°, and then rescaling so that e 1 ⋅ e 1 = 1 holds. Applying these rules, we find