Action of Lie Brackets on vector fields multiplied by functions

In summary, the product rule for applying the lie bracket or commutator to two vector fields involves the use of the product rule for derivatives on smooth functions. This can help with understanding the third term in the expression [fX+Z,Y] = f[X,Y] + [Z,Y] - (Yf)Xg.
  • #1
tut_einstein
31
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Hi,

Is there a specific product rule or something one must follow when applying the lie bracket/ commutator to two vector fields such that one of them is multiplied by a function and added to another vector field? This is the expression given in my textbook but I don't see how:

[fX+Z,Y] = f[X,Y] + [Z,Y] - (Yf)Xg

I don't see where the third term on the right hand side comes from.

I'd really appreciate some help on this because I'm self-learning differential geometry for a research project and almost all my doubts revolve around my not understanding how lie brackets work. So any help will be appreciated.

Thanks!
 
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  • #2
tut_einstein said:
Is there a specific product rule or something one must follow when applying the lie bracket/ commutator to two vector fields such that one of them is multiplied by a function

Yes.
tut_einstein said:
This is the expression given in my textbook

Which book?

Let's go back a couple of steps.

If [itex]X[/itex] is a vector field and [itex]f[/itex] and [itex]g[/itex] are smooth functions, then both [itex]Xf[/itex] and [itex]Xg[/itex] are functions. Because [itex]f[/itex] and [itex]g[/itex] are both functions, the product [itex]fg[/itex] is also a function on which [itex]X[/itex] can act. Consquently, [itex]X \left(fg\right)[/itex] is a function. [itex]X[/itex] acts like a derivative (is a derivation) on the set (ring) of smooth functions, i.e.,
[tex]X \left(fg\right) = g Xf + f Xg.[/tex]
Now, use the above and expand
[tex]\left[ fX,Y \right]g .[/tex]
 
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FAQ: Action of Lie Brackets on vector fields multiplied by functions

What is the definition of Lie brackets on vector fields multiplied by functions?

The Lie bracket of two vector fields multiplied by functions is a mathematical operation that calculates the commutator of two vector fields. It is defined as the difference between the vector fields multiplied by the functions evaluated at the same point.

How is the Lie bracket of two vector fields multiplied by functions calculated?

The Lie bracket of two vector fields multiplied by functions is calculated by taking the partial derivatives of the vector fields and functions, and then taking the difference between the two resulting vector fields multiplied by the functions evaluated at the same point.

What is the significance of Lie brackets on vector fields multiplied by functions in mathematics?

Lie brackets on vector fields multiplied by functions are important in the field of differential geometry as they are used to define the curvature of a manifold. They are also used in the study of dynamical systems and Lie algebras.

Can the Lie bracket of two vector fields multiplied by functions be extended to higher dimensions?

Yes, the concept of Lie brackets on vector fields multiplied by functions can be extended to higher dimensions, where the vector fields and functions are defined on a higher-dimensional manifold. The calculation and definition remain the same.

What are some real-world applications of Lie brackets on vector fields multiplied by functions?

Lie brackets on vector fields multiplied by functions have practical applications in physics, engineering, and economics. They are used in the study of fluid dynamics, control theory, and optimization problems, among others.

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