Search results
Results from the WOW.Com Content Network
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]
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 ...
Because the Rindler observers are vorticity-free, they are also hypersurface orthogonal. The orthogonal spatial hyperslices are t = t 0 {\displaystyle t=t_{0}} ; these appear as horizontal half-planes in the Rindler chart and as half-planes through T = X = 0 {\displaystyle T=X=0} in the Cartesian chart (see the figure above).
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 .
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 ...
Suppose the strong energy condition holds in some region of our spacetime, and let be a timelike geodesic unit vector field with vanishing vorticity, or equivalently, which is hypersurface orthogonal. For example, this situation can arise in studying the world lines of the dust particles in cosmological models which are exact dust solutions of ...
Here, saying that = is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that our spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime .