One-Form Fields

A one-form field (or covector field) assigns to each point a linear functional on tangent vectors. While a vector field eats a function and produces a number at each point, a one-form field eats a vector field and produces a scalar function.

The most natural one-form is the differential $df$ of a scalar function $f$:

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

This is the directional derivative of $f$ along $V$ — a purely functional definition.

Coordinate One-Forms

In coordinates $(x, y)$, the coordinate one-forms are $dx$ and $dy$:

$dx(V) = V^x = \text{the }x\text{-component of }V$ $dy(V) = V^y = \text{the }y\text{-component of }V$

These are the duals of the coordinate basis vectors $\partial/\partial x$ and $\partial/\partial y$.

Example

For $f(x,y) = x^2 + y^2$:

$df = 2x\,dx + 2y\,dy$

Acting on the constant vector field $V = \partial/\partial x$ at the point $(3,4)$:

$df(\partial/\partial x)(3,4) = 2 \cdot 3 = 6$

Your Task

Implement df(f) that returns the differential one-form of a scalar function $f$. The one-form should be callable as df(f)(V)(*point) where V(point) returns the vector components.