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  "body": "\n## Definition\n\nLet $\\gamma \\colon [a, b] \\to \\mathbb{R}^n$ be a [[rectifiable curve|0C28RT5BG7]] with image $C$, and let $F \\colon C \\to \\mathbb{R}^n$ be a continuous [[vector field|0N67NBKQ5T]].\nFor a partition $a = t_0 \u003c t_1 \u003c \\dots \u003c t_k = b$ and points $\\tau_i \\in [t_{i-1}, t_i]$, consider the sum $\\sum_{i=1}^{k} F(\\gamma(\\tau_i)) \\cdot \\big( \\gamma(t_i) - \\gamma(t_{i-1}) \\big)$.\nThe line integral of $F$ along $\\gamma$ is the limit of these sums as $\\max_i (t_i - t_{i-1}) \\to 0$. It is written\n$$\\int_\\gamma F \\cdot d\\mathbf{r} \\quad \\text{or} \\quad \\int_\\gamma F_1 \\, dx_1 + \\dots + F_n \\, dx_n.$$\n\n## Remarks\n\nThe limit exists, and it does not depend on the choice of the points $\\tau_i$.\nFor a $C^1$ curve, $\\int_\\gamma F \\cdot d\\mathbf{r} = \\int_a^b F(\\gamma(t)) \\cdot \\gamma'(t) \\, dt$, by the [[parametric formula|0W3W1VYWBP]].\nFor a regular curve with unit tangent vector $T$, $\\int_\\gamma F \\cdot d\\mathbf{r} = \\int_\\gamma (F \\cdot T) \\, ds$. So it is the line integral of the scalar field $F \\cdot T$.\nThe value depends on the direction of travel. [[Reversing the path|0FSCXYG856]] changes its sign.\n\n## Explanations\n\n### L1 human\n\nAt each small step along the curve, take the dot product of the field with the step.\nThis counts only the part of the field that pushes along the curve, and the line integral is the total of these contributions.\n",
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