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In differential topology, a mathematical discipline, and more specifically in Morse theory, a gradient-like vector field is a generalization of gradient vector field. The primary motivation is as a technical tool in the construction of Morse functions, to show that one can construct a function whose critical points are at distinct levels. One first constructs a Morse function, then uses gradient-like vector fields to move around the critical points, yielding a different Morse function. == Definition == Given a Morse function ''f'' on a manifold ''M,'' a gradient-like vector field ''X'' for the function ''f'' is, informally: * away from critical points, ''X'' points "in the same direction as" the gradient of ''f,'' and * near a critical point (in the neighborhood of a critical point), it equals the gradient of ''f,'' when ''f'' is written in standard form given in the Morse lemmas. Formally:〔(p. 63 )〕 * away from critical points, * around every critical point there is a neighborhood on which ''f'' is given as in the Morse lemmas: : and on which ''X'' equals the gradient of ''f.'' 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「Gradient-like vector field」の詳細全文を読む スポンサード リンク
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