## Question

Verify the claims made about the commutator of two vector fields at the end of section 2.3 (linearity, Leibnitz, component formula, transformation as a vector field).
The commutator of two vector fields X and Y is defined by its action on a function f(xμ) as
 [X,Y](f) ≡ X(Y(f))) - Y(X(f))) (2.20)

1) Linearity: if f and g are functions and a, b real numbers, the commutator is linear:
 [X,Y](af + bg) = a[X,Y](f) + b[X,Y](g) (2.21)
2) It obeys the Leibnitz rule:
 [X,Y](fg) = f[X,Y](g) + g[X,Y](f) (2.22)
3) The components of the vector field [X,Y]μ are given by:
 [X,Y]μ = Xλ∂λYμ - Yλ∂λXμ (2.23)
4) Transformation as a vector field:
Perform explicitly a coordinate transformation on the expression (2.23) to verify that all potentially non-tensorial pieces cancel and the result transforms like a vector field.

The first two were pretty easy. They just involved using the definition, using the fact that X and Y must be linear and / or obey the Leibnitz rule and then cancelling and / or using the definition again. My answers are almost line for line in agreement with the UCSB solution.

The third was a bit more difficult. I used the fact that the vector field is also a derivative operator. There were two slightly dubious steps in my proof: thaY(xλ )=Yλ and removing  entirely to get equation 0.5. But I think they are OK especially as they give the right answer!

I eventually understood the fourth question and knew where to start and where it should end. The four fairly hairy steps in between eluded me. I have followed the steps in detail in my pdf.

The UCSB solutions also proved the Jacobi identity for vector fields. It is

[ [X, Y ], Z ] + [ [Y, Z], X ] + [ [Z, X], Y ] = 0.
The proof was easy!