Vector Fields

A vector field assigns a tangent vector to every point of a manifold. In coordinates, a vector field $V$ has components $V^i(p)$ that vary smoothly from point to point.

A vector field acts as a differential operator on scalar functions:

$V[f](p) = \sum_i V^i(p) \frac{\partial f}{\partial x^i}(p)$

This produces a new scalar function — the directional derivative of $f$ along $V$.

Important Vector Fields on $\mathbb{R}^2$

Radial field: $X = x\partial/\partial x + y\partial/\partial y$

This field points radially outward. Acting on $f = x^2 + y^2$: $X[f](1,2) = x \cdot 2x + y \cdot 2y = 2 + 8 = 10$

Rotation field: $Y = -y\partial/\partial x + x\partial/\partial y$

This field rotates counterclockwise. Acting on $f = x^2 + y^2$ (which is rotationally symmetric): $Y[f](1,2) = -y \cdot 2x + x \cdot 2y = -4 + 4 = 0$

The rotation field kills radially-symmetric functions — it detects angular variation.

Your Task

Implement vector_field(comps_fn) where comps_fn(point) returns the component list $[V^0(p), V^1(p)]$. The result should be a function $V$ such that $V(f)(*point)$ gives the directional derivative.