Search Results for ""

The contravariant four-vector arising in special and general relativity, x^mu=[x^0; x^1; x^2; x^3]=[ct; x; y; z], (1) where c is the speed of light and t is time. ...

A change of basis is the transformation of coordinate-based vector and operator representations in a given vector space from one vector basis representation to another.

Given a vector bundle pi:E->M, its dual bundle is a vector bundle pi^*:E^*->M. The fiber bundle of E^* over a point p in M is the dual vector space to the fiber of E.

The normalized vector of X is a vector in the same direction but with norm (length) 1. It is denoted X^^ and given by X^^=(X)/(|X|), where |X| is the norm of X. It is also ...

On a Riemannian manifold M, tangent vectors can be moved along a path by parallel transport, which preserves vector addition and scalar multiplication. So a closed loop at a ...

An n×n complex matrix A is called positive definite if R[x^*Ax]>0 (1) for all nonzero complex vectors x in C^n, where x^* denotes the conjugate transpose of the vector x. In ...

Geometric objects lying in a common plane are said to be coplanar. Three noncollinear points determine a plane and so are trivially coplanar. Four points are coplanar iff the ...

Euclidean n-space, sometimes called Cartesian space or simply n-space, is the space of all n-tuples of real numbers, (x_1, x_2, ..., x_n). Such n-tuples are sometimes called ...

In a plane, consider a sum of N two-dimensional vectors with random orientations. Use phasor notation, and let the phase of each vector be random. Assume N unit steps are ...

Abstractly, the tensor direct product is the same as the vector space tensor product. However, it reflects an approach toward calculation using coordinates, and indices in ...