One-form

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A differential 1-form is a smooth section of the cotangent bundle; it assigns a covector to every point of a smooth manifold. In planar coordinates,

α = P (x, y) d x + Q (x, y) d y

v

P (x, y) v_{x} + Q (x, y) v_{y}

γ (t) = (x (t), y (t))

\int_{γ} α = \int_{a}^{b} [P (x (t), y (t)) x^{'} (t) + Q (x (t), y (t)) y^{'} (t)] d t .

Reversing the direction reverses the integral. A force expressed as a one-form therefore accumulates work along a directed path. In Euclidean coordinates its components can be displayed as arrows, but identifying a one-form with a vector field uses an inner product; the covector field itself does not require a Riemannian metric.