One-form

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 isReversing 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.

Let $W$ consist of real-linear maps $\alpha$ from smooth vector fields to smooth functions such that $X_p=0$ implies $\alpha (X)(p)=0$. These are exactly evaluations by smooth differential one-forms. Define ${\omega }_p(v)=\alpha (X)(p)$ using any global field with $X_p=v$; the vanishing condition makes it well-defined. Bump-extended local frame vectors make its coefficients smooth. Also $\alpha (fX)(p)=f(p)\alpha (X)(p)$, so real linearity and pointwise vanishing automatically give $C^{\infty }$-linearity. The evaluation correspondence is a natural module isomorphism.