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In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group into . In case is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted , is analytic and has as such a derivative , where is a path in the Lie algebra, and a closely related differential .〔 Appendix on analytic functions.〕 The formula for was first proved by Friedrich Schur (1891). It was later elaborated by Henri Poincaré (1899) in the context of the problem of expressing Lie group multiplication using Lie algebraic terms. It is also sometimes known as Duhamel's formula. The formula is important both in pure and applied mathematics. It enters into proofs of theorems such as the Baker–Campbell–Hausdorff formula, and it is used frequently in physics for example in quantum field theory, as in the Magnus expansion in perturbation theory, and in lattice gauge theory. Throughout, the notations and will be used interchangeably to denote the exponential given an argument, ''except'' when, where as noted, the notations have dedicated ''distinct'' meanings. The calculus-style notation is preferred here for better readability in equations. On the other hand, the -style is sometimes more convenient for inline equations, and is necessary on the rare occasions when there is a real distinction to be made. ==Statement== The derivative of the exponential map is given by〔 Theorem 5 Section 1.2〕 }}\frac. |cellpadding= 6 |border |border colour = #0073CF |bgcolor=#F9FFF7}} ;Explanation * is a (continuously differentiable) path in the Lie algebra with derivative . The argument is omitted where not needed. * is the linear transformation of the Lie algebra given by . It is the adjoint action of a Lie algebra on itself. *The fraction is given by the power series ::}} = \sum_^\infty \frac(\mathrm_X)^k. |}} derived from the power series of the exponential map of a linear endomorphism, as in matrix exponentiation〔 *When is a matrix Lie group, all occurrences of the exponential are given by their power series expansion. *When is ''not'' a matrix Lie group, is still given by its power series , while the other two occurrences of in the formula, which now are the exponential map in Lie theory, refer to the time-one flow of the left invariant vector field , i.e. element of the Lie algebra as defined in the general case, on the Lie group viewed as an analytic manifold. This still amounts to exactly the same formula as in the matrix case. *The formula applies to the case where is considered as a map on matrix space over or , see matrix exponential. When or , the notions coincide precisely. To compute the differential of at , , the standard recipe〔 : is employed. With the result〔 }}Y|}} follows immediately from . In particular, is the identity because (since is a vector space) and . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Derivative of the exponential map」の詳細全文を読む スポンサード リンク
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