Lie groups may be thought of as "multi-parameter" groups. Since 1-parameter groups of transformations can be studied infinitesimally giving rise to a vector field which infinitesimally generates it, it raises the natural question whether we can use vector fields to similarly study Lie groups infinitesimally. In this video which introduce the exponential map which realises this. Using left-invariant vector fields, we construct this exponential map from the tangent space to the Lie group (at the identity) to the Lie group itself.
