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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]
In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. An example is the quadric surface =
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 ...
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
(This is not true for example in the Boyer–Lindquist chart for the exterior region of the Kerr vacuum, where the timelike coordinate vector is not hypersurface orthogonal.) Note the last two fields are rotations of one-another, under the coordinate transformation ϕ ↦ ϕ + π / 2 {\displaystyle \phi \mapsto \phi +\pi /2} .
The gradient of a function is obtained by raising the index of the differential , whose components are given by: =; =; =, = = The divergence of a vector field with components is
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R 3 , the Gauss map is a map N : X → S 2 (where S 2 is the unit sphere ) such that for each p in X , the function value N ( p ) is ...
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 ...