One-form

Definition

Let $U \subseteq \mathbb{R}^n$ be open. A one-form on $U$ is a differential form of degree $1$, that is, a sum $\omega = f_1 \, dx_1 + \dots + f_n \, dx_n$ with functions $f_i \colon U \to \mathbb{R}$. At $x \in U$, it gives the linear map $\omega_x(v) = f_1(x) v_1 + \dots + f_n(x) v_n$ for $v \in \mathbb{R}^n$.

Remarks

For a differentiable function $g$ on $U$, the differential of $g$ is the one-form $dg = \sum_i \frac{\partial g}{\partial x_i} \, dx_i$. So $dg_x(v)$ is the directional derivative of $g$ at $x$ in direction $v$. For a continuous $\omega$ and a piecewise $C^1$ curve $\gamma \colon [a, b] \to U$, the integral is $\int_\gamma \omega = \int_a^b \omega_{\gamma(t)}\big( \gamma'(t) \big) \, dt$. This is the line integral of the vector field $(f_1, \dots, f_n)$. Under this correspondence, $dg$ corresponds to $\nabla g$, and $\omega$ is of the form $dg$ exactly when the field is conservative. On $\mathbb{R}^2 \setminus \{0\}$, the angle form $d\theta = \frac{-y \, dx + x \, dy}{x^2 + y^2}$ is a one-form. Its integral around a closed curve is $2\pi$ times the winding number of the curve about the origin.

Explanations

L1 human

A one-form is a rule that, at each point, turns a direction vector into a number, in a linear way. Integrating a one-form along a curve adds up these numbers for the velocity of the curve. It is the same thing as the work of a vector field.