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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]
Here () denotes the ()-orthogonal projection of onto the ()-orthogonal linear subspace to () (). Mean curvature of an immersion ... in the hypersurface case is
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 ...
A conformal map acting on a rectangular grid. Note that the orthogonality of the curved grid is retained. While vector operations and physical laws are normally easiest to derive in Cartesian coordinates, non-Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially boundary value problems, such as those arising in field theories of quantum ...
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.
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 .
The orthogonal spatial hyperslices are =; these appear as horizontal half-planes in the Rindler chart and as half-planes through = = in the Cartesian chart (see the figure above). Setting d t = 0 {\displaystyle dt=0} in the line element, we see that these have the ordinary Euclidean geometry, d σ 2 = d x 2 + d y 2 + d z 2 , ∀ x > 0 , ∀ y ...
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 ...