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Geometry index. In coordination chemistry and crystallography, the geometry index or structural parameter (τ) is a number ranging from 0 to 1 that indicates what the geometry of the coordination center is. The first such parameter for 5-coordinate compounds was developed in 1984. [1] Later, parameters for 4-coordinate compounds were developed. [2]
Principal symbol. The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature. The principal symbol of the map assigns to each a map from the space of symmetric (0,2)-tensors on to the space of (0,4)-tensors on given by.
The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r.
Solution techniques. Solving the geodesic equations means obtaining an exact solution, possibly even the general solution, of the geodesic equations. Most attacks secretly employ the point symmetry group of the system of geodesic equations. This often yields a result giving a family of solutions implicitly, but in many examples does yield the ...
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here ), and is denoted by the symbol . Given two linearly independent vectors a and b, the cross product, a × b ...
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas [1]) are fundamental formulas that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold.
Euclidean distance. In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names come from the ancient Greek ...
In linear algebra and functional analysis, the principal axis theorem is a geometrical counterpart of the spectral theorem. It has applications to the statistics of principal components analysis and the singular value decomposition. In physics, the theorem is fundamental to the studies of angular momentum and birefringence.