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  "body": "\n## Proof\n\nWe use Fubini's theorem and the fundamental theorem of calculus on each term.\n\n1. Since $F = (P, Q)$ is of [[class $C^1$|0TCBZMEYGN]] on an open set containing $R$, the [[partial derivatives|05AXG3QRKR]] $P_x = \\partial P / \\partial x$ and $Q_y = \\partial Q / \\partial y$ are continuous on $R$.\n2. By linearity of the [[double integral|0NKBMFC18R]], $\\iint_R (P_x + Q_y) \\, dA = \\iint_R P_x \\, dA + \\iint_R Q_y \\, dA$. This is the integral of the [[divergence|02NSMQVQWK]] of $F$.\n3. By [[Fubini's theorem|0R0T1VWH5X]] for the continuous function $P_x$, $\\iint_R P_x \\, dA = \\int_c^d \\left( \\int_a^b P_x(x, y) \\, dx \\right) dy$.\n4. Fix $y \\in [c, d]$. The function $x \\mapsto P(x, y)$ is continuous on $[a, b]$, and its derivative on $(a, b)$ is the continuous function $x \\mapsto P_x(x, y)$.\n5. By the [[fundamental theorem of calculus|0NHGY0N0JB]], $\\int_a^b P_x(x, y) \\, dx = P(b, y) - P(a, y)$. So $\\iint_R P_x \\, dA = \\int_c^d \\big( P(b, y) - P(a, y) \\big) \\, dy$.\n6. By Fubini's theorem in the other order, $\\iint_R Q_y \\, dA = \\int_a^b \\left( \\int_c^d Q_y(x, y) \\, dy \\right) dx$.\n7. For fixed $x \\in [a, b]$, the fundamental theorem of calculus gives $\\int_c^d Q_y(x, y) \\, dy = Q(x, d) - Q(x, c)$. So $\\iint_R Q_y \\, dA = \\int_a^b \\big( Q(x, d) - Q(x, c) \\big) \\, dx$.\n8. Adding steps 5 and 7 with step 2 gives the stated formula.\n\n## Idea\n\nOn a rectangle, each part of the divergence can be integrated first in its own variable. The fundamental theorem of calculus turns that inner integral into a difference of boundary values. So the integral of $P_x$ only sees the right and left sides, and the integral of $Q_y$ only sees the top and bottom sides.\n",
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