= One-form
{wiki}
= Differential 1-form
{synonym}
= Covector field
{synonym}
= Differential one-form
{synonym}
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, $\alpha=P(x,y)\,dx+Q(x,y)\,dy$ evaluates a <tangent vector> $v$ as $P(x,y)v_x+Q(x,y)v_y$.
Along a <parametric curve> $\gamma(t)=(x(t),y(t))$, its <line integral> is
$$
\int_\gamma\alpha=\int_a^b\left[P(x(t),y(t))x'(t)+Q(x(t),y(t))y'(t)\right]dt.
$$
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>.
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