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Living_Dog
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Try this thread: https://www.physicsforums.com/showthread.php?t=150200Terilien said:
Try this thread: https://www.physicsforums.com/showthread.php?t=150200Terilien said:i was curious as to what exactly this is and more importantly, what motivates it. what are its applications?
A Lie Derivative is a mathematical operation used to measure the change of a vector field along the flow of another vector field. It can be thought of as a way to measure how much a vector field is changing in a specific direction.
The Lie Derivative is calculated using the Lie bracket, which is a commutator of two vector fields. It involves taking the partial derivative of one vector field with respect to the other and subtracting the partial derivative of the second vector field with respect to the first.
Lie Derivatives have various applications in mathematics and physics, particularly in the fields of differential geometry and general relativity. They are used to study the behavior of vector fields and can also be used to define symmetries in physical systems.
Lie Derivatives are closely related to Lie Groups, which are mathematical objects that describe continuous symmetries in a system. The Lie Derivative can be used to define the action of a Lie Group on a vector field, which can then be extended to other mathematical objects.
Yes, Lie Derivatives can be generalized to higher dimensions and are often used in the study of manifolds. In higher dimensions, the Lie Derivative is defined as the directional derivative of a tensor field along the flow of a vector field, rather than just a vector field.