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  "body": "\n## Summary\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] with Lie algebra $\\mathfrak{g} = T_eG$.\nThe exponential map sends $X \\in \\mathfrak{g}$ to the value at time $1$ of the [[one-parameter subgroup|YHA64AP1HN]] with initial velocity $X$.\nFor matrix groups it is the [[matrix exponential|020M04FN1H]].\nIt restricts to a diffeomorphism from a neighbourhood of $0$ onto a neighbourhood of the identity.\nSo it carries information from the Lie algebra to the group.\nIt is the main tool for passing between Lie groups and Lie algebras, and it differs in general from the [[exponential map|Y7Z0AGY1A7]] of Riemannian geometry.\n\n## Explanations\n\n### L1 human\n\nStart at the identity of the group and move with a constant velocity $X$, in the sense that the motion is a one-parameter subgroup.\nWhere you are after one unit of time is $\\exp(X)$.\nOn the circle group, with velocity $\\theta$, you end at the point $e^{i\\theta}$, which is the ordinary exponential function.\nFor a square matrix $A$, $\\exp(A)$ is the series $I + A + A^2/2! + A^3/3! + \\cdots$.\n",
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