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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open.\nA one-form on $U$ is a [[differential form|0QHTX7WQEY]] of degree $1$, that is, a sum $\\omega = f_1 \\, dx_1 + \\dots + f_n \\, dx_n$ with functions $f_i \\colon U \\to \\mathbb{R}$.\nAt $x \\in U$, it gives the linear map $\\omega_x(v) = f_1(x) v_1 + \\dots + f_n(x) v_n$ for $v \\in \\mathbb{R}^n$.\n\n## Remarks\n\nFor 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 $v$.\nFor 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|0YS8RX87F4]] of the [[vector field|0N67NBKQ5T]] $(f_1, \\dots, f_n)$.\nUnder this correspondence, $dg$ corresponds to $\\nabla g$, and $\\omega$ is of the form $dg$ exactly when the field is [[conservative|05R9NF2J49]].\nOn $\\mathbb{R}^2 \\setminus \\{0\\}$, 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.\n\n## Explanations\n\n### L1 human\n\nA one-form is a rule that, at each point, turns a direction vector into a number, in a linear way.\nIntegrating 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.\n",
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