# Question on the notaion used to define Lie Derviative

• logarithmic
In summary, the notation used to define Lie Derivative is a subscripted "L" followed by a vector field inside parentheses, acting on a tensor or differential form. It differs from traditional derivatives by measuring the change of a tensor or differential form along the flow of a vector field, and its geometric interpretation is as the rate of change of the tensor or form as it moves and rotates along the curves defined by the vector field. Lie Derivative can be extended to higher dimensions and has practical applications in physics, such as in general relativity, fluid mechanics, and differential geometry.
logarithmic
I have a definition of the Lie derivative, that is the one found here: http://planetmath.org/encyclopedia/LieDerivative2.html

However, I'm not sure what the notation $$Y_{\theta_t(p)}$$ used in that article means.

Y is a vector field and $$\theta_t(p)$$ is a function. Does it mean evaluate Y at \theta_t(p)}[/tex]?

Can someone explain.

Last edited by a moderator:
logarithmic said:
Does it mean evaluate Y at $$\theta_t(p)}$$?

## 1. What is the notation used to define Lie Derivative?

The notation used to define Lie Derivative is a subscripted "L" followed by a vector field inside parentheses, acting on a tensor or differential form. It can also be represented using mathematical symbols such as [L_X, Y] or L_X(Y).

## 2. How is Lie Derivative different from traditional derivatives?

Lie Derivative is different from traditional derivatives because it measures the change of a tensor or differential form along the flow of a vector field, rather than simply at a point. This means that Lie Derivative takes into account the movement and transformation of the tensor or form, rather than just its value at a specific point.

## 3. What is the geometric interpretation of Lie Derivative?

The geometric interpretation of Lie Derivative is that it measures how a tensor or differential form changes along the flow of a vector field. This can be thought of as the rate of change of the tensor or form as it moves and rotates along the curves defined by the vector field.

## 4. Can Lie Derivative be extended to higher dimensions?

Yes, Lie Derivative can be extended to higher dimensions by applying the same definition and notation to tensors and differential forms in higher dimensions. This allows for the calculation of Lie Derivatives in spaces with multiple dimensions, such as in general relativity.

## 5. What are some practical applications of Lie Derivative in physics?

Lie Derivative is commonly used in the study of general relativity, where it is used to describe the motion and transformation of tensors and forms in curved spacetime. It is also used in fluid mechanics to study the movement and deformation of fluids, and in differential geometry to study symmetries and transformations of manifolds.

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