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The metric g can take up to two vectors or vector fields X, Y as arguments. In the former case the output is a number, the (pseudo-)inner product of X and Y. In the latter case, the inner product of X p, Y p is taken at all points p on the manifold so that g(X, Y) defines a smooth function on M. Vector fields act (by definition) as differential ...
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3) , the group of all rotation matrices ...
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
[10] [11] Carl Friedrich Gauss had discovered quaternions in 1819, but this work was not published until 1900. [ 12 ] [ 13 ] Hamilton knew that the complex numbers could be interpreted as points in a plane , and he was looking for a way to do the same for points in three-dimensional space .
Let P and Q be two sets, each containing N points in .We want to find the transformation from Q to P.For simplicity, we will consider the three-dimensional case (=).The sets P and Q can each be represented by N × 3 matrices with the first row containing the coordinates of the first point, the second row containing the coordinates of the second point, and so on, as shown in this matrix:
In a 3-dimensional projective space , let L be a line through distinct points x and y with homogeneous coordinates (x 0 : x 1 : x 2 : x 3) and (y 0 : y 1 : y 2 : y 3). The Plücker coordinates p ij are defined as follows:
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 ...