Tangent Vectors as Derivations

Traditional calculus defines a tangent vector as an arrow at a point. But this geometric picture breaks down on abstract manifolds. The book takes the algebraic approach: a tangent vector is defined by how it acts on functions.

A tangent vector $v$ at a point $p$ is a derivation — a linear map from smooth functions to real numbers:

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

where $v^0, v^1, \ldots$ are the components of $v$ in the coordinate basis.

The Coordinate Basis

For rectangular coordinates $(x, y)$, the coordinate basis vectors are: $\partial/\partial x: \quad f \mapsto \frac{\partial f}{\partial x}(p)$ $\partial/\partial y: \quad f \mapsto \frac{\partial f}{\partial y}(p)$

These are the "unit vectors" in each coordinate direction, expressed as operators on functions.

Example

The vector $v = [3, -1]$ (meaning $3\partial/\partial x - \partial/\partial y$) acts on $f(x,y) = x^2 + y^2$:

$v[f](1, 2) = 3 \cdot \frac{\partial}{\partial x}(x^2+y^2)\Big|_{(1,2)} + (-1) \cdot \frac{\partial}{\partial y}(x^2+y^2)\Big|_{(1,2)}$ $= 3 \cdot 2 + (-1) \cdot 4 = 2$

Your Task

Implement tangent_vector(components) that returns a function v such that v(f)(*point) computes the directional derivative of f in direction components at point.