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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.
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 (,).
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 ...
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.
Positive space refers to the areas of the work with a subject, while negative space is the space without a subject. [6] Open and closed space coincides with three-dimensional art, like sculptures, where open spaces are empty, and closed spaces contain physical sculptural elements.
For example, the image of a curve can cover a square in the plane (space-filling curve), and a simple curve may have a positive area. [10] Fractal curves can have properties that are strange for the common sense. For example, a fractal curve can have a Hausdorff dimension bigger than one (see Koch snowflake) and even a positive area. An example ...
The oriented area of any polygon can be written as a signed real number coefficient (the signed area of the shape) times the oriented area of a designated polygon declared to have unit area; in the case of the Euclidean plane, this is typically a unit square.
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 ...