The Lie bracket.

One use of this discussion is the definition of the Lie bracket of two vector fields. Let $X$ and $Y$ be two vector fields and write them locally as

$$X = \sum_{i=1}^n X^i \frac{\partial\phantom{\psi^i}}{\psi^i}$$

and

$$Y = \sum_{i=1}^n Y^i \frac{\partial\phantom{\psi^i}}{\psi^i}.$$

so that

Then define

$$[X, Y] = \sum_{i, j=1}^n (X^j\frac{\partial Y^i}{\partial \psi^j}... ...\frac{\partial X^i}{\partial \psi^j}) \frac{\partial\phantom{\psi^i}}{\psi^i}.$$

We leave it as an exercise to show that this transforms as a vector field. We call $[X, Y]$ the Lie bracket of the vector fields $X$ and $Y$. Lie is named after Sophus Lie and pronounced "lee".