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In vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector fields. They can be characterized in a number of different ways, many of which involve the curl.
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in an ambient space of dimension n, generally a Euclidean space, an affine space or a projective space. [1]
Google Earth images are of this type. The document (digital or paper) representing an orthophotomosaic with additional marginal information like a title, north arrow, scale bar and cartographical information is called an orthophotomap or image map. Often these maps show additional point, line or polygon layers (like a traditional map) on top of ...
Considered extrinsically, as a hypersurface embedded in (+) -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the radius) from a given center point. Its interior , consisting of all points closer to the center than the radius, is an ( n + 1 ) {\displaystyle (n+1)} -dimensional ball .
Rings of constant ξ 1 and ξ 2 above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when η equals 0 or π / 2 , these coordinates describe a circle. The round metric on the 3-sphere in these coordinates is given by
In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V.The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can ...
A smooth quadric over a field k is a projective homogeneous variety for the orthogonal group (and for the special orthogonal group), viewed as linear algebraic groups over k. Like any projective homogeneous variety for a split reductive group, a split quadric X has an algebraic cell decomposition, known as the Bruhat decomposition. (In ...
The Gauss map can be defined for hypersurfaces in R n as a map from a hypersurface to the unit sphere S n − 1 ⊆ R n. For a general oriented k-submanifold of R n the Gauss map can also be defined, and its target space is the oriented Grassmannian ~,, i.e. the set of all oriented k-planes in R n. In this case a point on the submanifold is ...