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The Definition of Lie Derivative

Published online by Cambridge University Press:  20 January 2009

T. J. Willmore
Mathematical Institute, University of Liverpool
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The differential operation known as Lie derivation was introduced by W. Slebodzinski in 1931, and since then it has been used by numerous investigators in applications in pure and applied mathematics and also in physics. A recent monograph by Kentaro Yano (2) devoted to the theory and application of Lie derivatives gives some idea of the wide range of its uses. However, in this monograph, as indeed in other treatments of the subject, the Lie derivative of a tensor field is defined by means of a formula involving partial derivatives of the given tensor field. It is then proved that the Lie derivative is a differential invariant, i.e. it is independent of a transformation from one allowable coordinate system to another. Sometimes some geometrical motivation is given in explanation of the formula, but this is seldom very satisfying.

Research Article
Copyright © Edinburgh Mathematical Society 1960



(1)Willmore, T. J., Generalised torsional derivation, Atti. Accad. Naz. Lincei, 26 (1959), 649653.Google Scholar
(2)Kentaro, Yano, The Theory of Lie Derivatives and its Applications, North-Holland Publishing Co., Groningen (1957).Google Scholar