Generator (mathematics)

A list of examples of generating sets follow.

In the study of differential equations, and commonly those occurring in physics, one has the idea of a set of infinitesimal displacements that can be extended to obtain a manifold, or at least, a local part of it, by means of integration.

The general concept is of using the exponential map to take the vectors in the tangent space and extend them, as geodesics, to an open set surrounding the tangent point.

In this case, it is not unusual to call the elements of the tangent space the generators of the manifold.

When the manifold possesses some sort of symmetry, there is also the related notion of a charge or current, which is sometimes also called the generator, although, strictly speaking, charges are not elements of the tangent space.

The 5th roots of unity in the complex plane under multiplication form a group of order 5. Each non-identity element by itself is a generator for the whole group.