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  "body": "\n## Statement\n\nLet $\\gamma \\colon [a, b] \\to \\mathbb{R}^n$ be a $C^1$ curve with image $C$.\nLet $f \\colon C \\to \\mathbb{R}$ be continuous, and let $F \\colon C \\to \\mathbb{R}^n$ be a continuous vector field.\nThen\n$$\\int_\\gamma f \\, ds = \\int_a^b f(\\gamma(t)) \\, \\|\\gamma'(t)\\| \\, dt, \\qquad \\int_\\gamma F \\cdot d\\mathbf{r} = \\int_a^b F(\\gamma(t)) \\cdot \\gamma'(t) \\, dt.$$\n\n## Remarks\n\nThe left sides are the [[line integral of a scalar field|0M50E4WYWJ]] and the [[line integral of a vector field|0YS8RX87F4]]. The right sides are ordinary integrals in one variable.\nThe formula follows from $\\Delta s_i = \\int_{t_{i-1}}^{t_i} \\|\\gamma'(t)\\| \\, dt$, the mean value theorem for each component of $\\gamma$, and the uniform continuity of $f \\circ \\gamma$ and $\\gamma'$.\nThe formulas extend to piecewise $C^1$ curves, by summing over the pieces.\nExample: for $F(x, y) = (-y, x)$ and $\\gamma(t) = (\\cos t, \\sin t)$, $t \\in [0, 2\\pi]$, the integrand is $\\sin^2 t + \\cos^2 t = 1$, so $\\int_\\gamma F \\cdot d\\mathbf{r} = 2\\pi$.\n\n## Explanations\n\n### L1 human\n\nTo compute a line integral, write the curve with a parameter $t$.\nThen $ds$ becomes $\\|\\gamma'(t)\\| \\, dt$ and $d\\mathbf{r}$ becomes $\\gamma'(t) \\, dt$, and the line integral is an ordinary integral in $t$.\n",
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