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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 ...
Lemma 1 (Decomposition Lemma) — Assume is a rectifiable, positively oriented Jordan curve in the plane and let be its inner region. For every positive real δ {\displaystyle \delta } , let F ( δ ) {\displaystyle {\mathcal {F}}(\delta )} denote the collection of squares in the plane bounded by the lines x = m δ , y = m δ {\displaystyle x=m ...
For example, R n has a standard volume form given by dx 1 ∧ ⋯ ∧ dx n. Given a volume form on M, the collection of all charts U → R n for which the standard volume form pulls back to a positive multiple of ω is an oriented atlas. The existence of a volume form is therefore equivalent to orientability of the manifold.
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 ...
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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
Science & Tech. Shopping. Sports. Weather. 24/7 Help. For premium support please call: 800-290-4726 more ways to reach us. ... Footage of her bounding off to the animal is positively charming.