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  "body": "\n## Proof\n\nWe pull both integrals back to $D$ and apply Green's theorem there.\n\n1. Let $r$ be $C^2$ on an open set $W \\supseteq D$, and let $F = (F_1, F_2, F_3)$ be $C^1$ on an open set $V \\supseteq S$. The set $W' = W \\cap r^{-1}(V)$ is open, since $r$ is continuous, and it contains $D$.\n2. Write $r_u, r_v$ for the partial derivatives of the [[parametrised surface|05788J51WZ]] $r$, and $a = r_u$, $b = r_v$ with components $a_j, b_j$. Write $\\partial_j F_i = \\partial F_i / \\partial x_j$.\n3. On $W'$ define $L(u, v) = F(r(u, v)) \\cdot r_u(u, v)$ and $M(u, v) = F(r(u, v)) \\cdot r_v(u, v)$.\n4. Each $F_i$ is differentiable, by [[continuous partial derivatives imply differentiability|0KXSK8TEEB]]. By the [[chain rule|0EAAS2EHDR]], $\\partial_v (F_i \\circ r) = \\sum_j \\partial_j F_i(r) \\, b_j$ and $\\partial_u (F_i \\circ r) = \\sum_j \\partial_j F_i(r) \\, a_j$.\n5. By the [[product rule|0V9A43H7WV]] in each variable, $\\partial_v L = \\sum_{i, j} \\partial_j F_i(r) \\, b_j a_i + F(r) \\cdot r_{uv}$ and $\\partial_u M = \\sum_{i, j} \\partial_j F_i(r) \\, a_j b_i + F(r) \\cdot r_{vu}$.\n6. These partial derivatives are continuous on $W'$, because $F$ is $C^1$ and $r$ is [[$C^2$|0TCBZMEYGN]]. By [[Schwarz's theorem|0NNQY8WCW6]], $r_{uv} = r_{vu}$.\n7. Hence $\\partial_u M - \\partial_v L = \\sum_{i, j} \\partial_j F_i(r) \\, (a_j b_i - a_i b_j)$. The terms with $i = j$ are $0$. Grouping the terms $(i, j)$ and $(j, i)$, the sum is $\\sum_{i \u003c j} (\\partial_j F_i - \\partial_i F_j)(r) \\, (a_j b_i - a_i b_j)$.\n8. By the definitions of the [[curl|0GDNHS9J2X]] and the [[cross product|0SEDGXG9S3]], the pair $(1, 2)$ gives $(\\partial_1 F_2 - \\partial_2 F_1)(a_1 b_2 - a_2 b_1) = (\\nabla \\times F)_3 \\, (a \\times b)_3$.\n9. The pair $(1, 3)$ gives $(\\partial_3 F_1 - \\partial_1 F_3)(a_3 b_1 - a_1 b_3) = (\\nabla \\times F)_2 \\, (a \\times b)_2$.\n10. The pair $(2, 3)$ gives $(\\partial_2 F_3 - \\partial_3 F_2)(a_2 b_3 - a_3 b_2) = (\\nabla \\times F)_1 \\, (a \\times b)_1$.\n11. By steps 7 to 10, $\\partial_u M - \\partial_v L = (\\nabla \\times F)(r) \\cdot (r_u \\times r_v)$ on $W'$.\n12. Let $\\gamma = (\\gamma_1, \\gamma_2) \\colon [\\alpha, \\beta] \\to \\R^2$, with break points $\\alpha = p_0 \u003c \\dots \u003c p_m = \\beta$ such that $\\gamma$ is $C^1$ on each $[p_{l-1}, p_l]$.\n13. On $(p_{l-1}, p_l)$, the [[chain rule along a curve|0A8TJ0VHC7]] gives $(r \\circ \\gamma)'(t) = r_u(\\gamma(t)) \\, \\gamma_1'(t) + r_v(\\gamma(t)) \\, \\gamma_2'(t)$. The right side is continuous on $[p_{l-1}, p_l]$.\n14. Apply the [[mean value theorem|0BARQY52N5]] in each component. The one-sided derivatives of $r \\circ \\gamma$ at $p_{l-1}$ and $p_l$ exist and equal the limits of the right side. So $r \\circ \\gamma$ is $C^1$ on each piece, and $\\partial S = r \\circ \\gamma$ is piecewise $C^1$.\n15. By [[additivity|00ZEBSMNNV]] and the [[parametric formula|0W3W1VYWBP]] on each piece, $\\oint_{\\partial S} F \\cdot d\\mathbf{r} = \\sum_l \\int_{p_{l-1}}^{p_l} F(r(\\gamma(t))) \\cdot (r \\circ \\gamma)'(t) \\, dt$.\n16. By step 13 and the definition of $L$ and $M$, the integrand is $L(\\gamma(t)) \\, \\gamma_1'(t) + M(\\gamma(t)) \\, \\gamma_2'(t)$. With the parametric formula and additivity again, $\\oint_{\\partial S} F \\cdot d\\mathbf{r} = \\oint_\\gamma (L \\, du + M \\, dv)$, a [[line integral|0YS8RX87F4]] in the plane.\n17. The functions $L$ and $M$ have continuous partial derivatives on the open set $W' \\supseteq D$, by step 6. By [[Green's theorem|0HR3N02317]] and step 11, $\\oint_\\gamma (L \\, du + M \\, dv) = \\iint_D (\\nabla \\times F)(r) \\cdot (r_u \\times r_v) \\, du \\, dv$.\n18. Since $r$ is regular, $r_u \\times r_v \\ne 0$ on $D$, and $n = (r_u \\times r_v) / \\norm{r_u \\times r_v}$. By the definitions of the [[flux|0YPRY7T4PK]] and the [[surface integral|0DQ8ERJZNS]], $\\iint_S (\\nabla \\times F) \\cdot n \\, dS = \\iint_D (\\nabla \\times F)(r) \\cdot (r_u \\times r_v) \\, du \\, dv$.\n19. Steps 16 to 18 give $\\oint_{\\partial S} F \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times F) \\cdot n \\, dS$.\n\n## Idea\n\nThrough the parametrisation, the circulation of $F$ around $\\partial S$ becomes a line integral of the plane field $(L, M) = (F \\cdot r_u, F \\cdot r_v)$ around $\\gamma$. Green's theorem turns it into a double integral of $\\partial_u M - \\partial_v L$. The second derivatives of $r$ cancel, and what remains is exactly the curl of $F$ dotted with the normal vector $r_u \\times r_v$.\n",
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