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sit.think.solve
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Loosely speaking or Intuitively how should one understand the difference between Lie Derivative and Covariant derivative? Definitions for both sounds awfully similar...
A Lie derivative is a measure of how a tensor field changes along the flow of a vector field, while a covariant derivative is a measure of how a tensor field changes along a specified direction in a curved space.
Lie derivatives are a special case of covariant derivatives, where the direction of change is given by the vector field generating the flow. In other words, the Lie derivative along a vector field is equivalent to the covariant derivative along that same vector field.
Imagine a river flowing along a curved path. The Lie derivative measures how a buoy floating in the river moves along with the flow, while the covariant derivative measures how the buoy moves relative to the river bank. Both are measures of how an object changes as it moves along a path.
Lie and covariant derivatives are used extensively in differential geometry, general relativity, and theoretical physics. They are essential tools for understanding the behavior of objects in curved spaces and play a crucial role in the mathematics behind Einstein's theory of gravity.
Lie derivatives can be calculated using the Lie bracket, which describes how two vector fields change with respect to each other. Covariant derivatives, on the other hand, require knowledge of the connection coefficients, which describe how the basis vectors of a curved space change as one moves along a given direction. Both can be calculated using various mathematical techniques, such as the exterior derivative or coordinate transformations.