Search results
Results from the WOW.Com Content Network
exponential map (Lie theory) from a Lie algebra to a Lie group, More generally, in a manifold with an affine connection, (), where is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with respect to appropriate affine connections.
The ordinary exponential function of mathematical analysis is a special case of the exponential map when is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however ...
The exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography. In Riemannian geometry, an exponential map is a map from a subset of a tangent space T p M of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical ...
- The graph of an exponential function in polar coordinates is a logarithmic spiral or equiangular spiral. [17] [18] In a logarithmic spiral with pitch angle 45 o the length of a radius vector increases by a factor when the polar angle increases by one radian. And by the factor at a 180 o switch.
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.
In case G is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted exp:g → G, is analytic and has as such a derivative d / dt exp(X(t)):Tg → TG, where X(t) is a C 1 path in the Lie algebra, and a closely related differential dexp:Tg → TG. [2] The formula for dexp was first proved ...
Exponential function; Exponential generating function; Exponential-Golomb coding; Exponential growth; Exponential hierarchy; Exponential integral; Exponential integrator; Exponential map (Lie theory) Exponential map (Riemannian geometry) Exponential map (discrete dynamical systems) Exponential notation; Exponential object (category theory ...
The exponential map is a mapping from the tangent space at p to M: : which is a diffeomorphism in a neighborhood of zero. Gauss' lemma asserts that the image of a sphere of sufficiently small radius in T p M under the exponential map is perpendicular to all geodesics originating at p.