Expansion of the commutator of two vector fields

In summary, the conversation discusses the expansion of the commutator of two vector fields and the confusion about a specific term in the expansion. The response explains that the term can be obtained through the product rule and that the commutator is bilinear.
  • #1
tut_einstein
31
0
Hi,

I don't understand a particular coordinate expansion of the commutator of 2 vector fields:

[X, Y ]f = X(Y f) − Y (Xf) = X_be_b(Y _ae_af) − Y _be_b(X_ae_af)
= (X_b(e_bY_ a) − Y _b(e_bX_a))e_af + X_aY _b[e_a, e_b]f


X,Y = Vector fields
f = function

X_i = Components of X and same for Y
e_i = coordinates of the vector space

I don't understand how to get the third term in the 2nd line. I can tell that it's probably a product rule but I don't see how to get it.

Thanks!
 
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  • #2
If we write X= (sum)X_i d/dxi &c., we only have to notice that the commutator is bilinear & the terms follow.
 

Related to Expansion of the commutator of two vector fields

1. What is a commutator of two vector fields?

A commutator of two vector fields is a mathematical operation that combines two vector fields and measures their non-commutativity, or the extent to which the order of the fields affects the result. It is denoted by [V,W] and is defined as the difference between the vector field obtained by applying V to W and the vector field obtained by applying W to V.

2. Why is the expansion of the commutator of two vector fields important?

The expansion of the commutator of two vector fields is important because it helps us understand the behavior of vector fields and their interactions. It can also provide insight into the underlying symmetries and properties of a system.

3. How is the commutator of two vector fields expanded?

The commutator of two vector fields can be expanded using the Jacobi identity, which states that [V,[W,Z]] + [W,[Z,V]] + [Z,[V,W]] = 0. This identity allows us to simplify the expansion and express it in terms of the Lie bracket, which is a fundamental operation in differential geometry.

4. Can the expansion of the commutator of two vector fields be used to solve differential equations?

Yes, the expansion of the commutator of two vector fields can be used to solve differential equations. It allows us to rewrite the original equation in terms of the Lie bracket, which can then be solved using various techniques such as power series or numerical methods.

5. Are there any real-world applications of the expansion of the commutator of two vector fields?

Yes, the expansion of the commutator of two vector fields has many real-world applications in fields such as physics, engineering, and computer science. It is commonly used in the study of fluid mechanics, electromagnetism, and quantum mechanics, among others. It also has practical applications in control systems, robotics, and machine learning algorithms.

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