Exploring the Function and Applications of Lie Derivative

Thread starter Terilien
Start date Apr 16, 2007
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Derivative Lie derivative

In summary, according to the author, the Lie derivative is a derivative that is always in the tangent space to the manifold. This derivative is useful for two reasons: first, it is a second derivative, and second, it is a derivative that measures the change in a vector field with respect to what it would have been had it been transported by the flow of the vector field.

Jun 13, 2010

Living_Dog

Terilien said:

i was curious as to what exactly this is and more importantly, what motivates it. what are its applications?

Try this thread: https://www.physicsforums.com/showthread.php?t=150200

<h2>1. What is a Lie Derivative?</h2><p>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.</p><h2>2. How is the Lie Derivative calculated?</h2><p>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.</p><h2>3. What are the applications of Lie Derivatives?</h2><p>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.</p><h2>4. How does the Lie Derivative relate to Lie Groups?</h2><p>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.</p><h2>5. Can Lie Derivatives be generalized to higher dimensions?</h2><p>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.</p>

1. What is a Lie Derivative?

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.

2. How is the Lie Derivative calculated?

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.

3. What are the applications of Lie Derivatives?

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.

4. How does the Lie Derivative relate to Lie Groups?

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.

5. Can Lie Derivatives be generalized to higher dimensions?

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.