The Lie Bracket

Two vector fields $V$ and $W$ can be applied in sequence to a function $f$: first apply $W$ to get $W[f]$, then apply $V$ to get $V[W[f]]$. In general, the result depends on the order:

$V[W[f]] \neq W[V[f]]$

The Lie bracket $[V, W]$ measures this failure to commute:

$[V, W][f] = V[W[f]] - W[V[f]]$

The Lie bracket is itself a vector field! This is a fundamental result: the space of vector fields is closed under the Lie bracket.

In Coordinates

For $V = V^i \partial_i$ and $W = W^j \partial_j$:

$[V, W]^k = V^i \partial_i W^k - W^i \partial_i V^k$

Geometric Meaning

The Lie bracket measures the obstruction to commutativity of flows. If you flow along $V$ for time $\epsilon$, then along $W$ for $\epsilon$, then back along $V$ for $\epsilon$, then back along $W$ for $\epsilon$, you end up displaced by $\epsilon^2 [V, W]$ from where you started.

Examples

Your Task

Implement lie_bracket(V, W) where $V$ and $W$ are vector fields (as built by vector_field). Return the Lie bracket $[V, W]$ as a vector field.