- #1
Silviu
- 624
- 11
Hello! So I have 2 vector fields on a manifold ##X=X^\mu\frac{\partial}{\partial x^\mu}## and ##Y=Y^\mu\frac{\partial}{\partial x^\mu}## and this statement: "Neither XY nor YX is a vector field since they are second-order derivatives, however ##[X, Y]## is a vector field". Intuitively makes sense but I am not sure how to show it mathematically. I tried this: for a function f defined on M, ##[X,Y]f = X[Y[f]]-Y[X[f]]=X^\mu\frac{\partial}{\partial x^\mu}[Y^\nu \frac{\partial}{\partial x^\nu}f]-Y^\mu \frac{\partial}{\partial x^\mu}[X^\nu \frac{\partial}{\partial x^\nu}f]##. It kinda makes sense that you first apply the vector on the right to f, which gives you a number, and the vector on the right remains just as an operator, and hence the whole stuff is a vector (but I am not sure if the results on the right can be put before the derivative on the left, such that we have an actual vector). For the other case ##XY[f]## I am not really sure how to combine the indices and what to get outside the derivatives. I guess I should have something like ##X^\mu Y^\nu \frac{\partial}{\partial x^\mu x^\nu}##, I think, which is not a vector (would it be a tensor?), but I am not sure. Can someone help me a bit here? Thank you!