View source for Lie differential

''of a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l0585701.png" /> in the direction of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l0585702.png" />''

The principal linear part of increment of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l0585703.png" /> under its transformation induced by the local one-parameter group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l0585704.png" /> of transformations of the manifold generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l0585705.png" />. The Lie differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l0585706.png" /> of a tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l0585707.png" /> in the direction of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l0585708.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l0585709.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l05857010.png" /> is the [[Lie derivative|Lie derivative]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l05857011.png" /> in the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l05857012.png" />.

The concept of a Lie differential admits the following physical interpretation. If a one-parameter transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l05857013.png" /> of a domain of the Euclidean space describes the stationary flow of a liquid with velocity field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l05857014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l05857015.png" /> is time and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l05857016.png" /> is a tensor field that describes some characteristic of the liquid (the deformation velocity tensor, stress tensor, density, etc.), then the Lie differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l05857017.png" /> describes the principal linear part of variation with time of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058570/l05857018.png" /> from the point of view of an observer moving with the liquid, that is, in Lagrange variables.