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  "body": "\n## Statement\n\nLet $a \u003c b$ and $c \u003c d$, and let $R = [a, b] \\times [c, d]$.\nLet $F = (P, Q)$ be a vector field of class $C^1$ on an open set that contains $R$.\nThen\n$$\\iint_R \\left( \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} \\right) dA = \\int_c^d \\big( P(b, y) - P(a, y) \\big) \\, dy + \\int_a^b \\big( Q(x, d) - Q(x, c) \\big) \\, dx.$$\n\n## Remarks\n\nThe right side is the outward flux $\\oint_{\\partial R} F \\cdot n \\, ds$. Here the outward unit normal $n$ is $(1, 0)$, $(-1, 0)$, $(0, 1)$ and $(0, -1)$ on the right, left, top and bottom sides.\nProof: by [[Fubini's theorem|0R0T1VWH5X]], $\\iint_R \\partial P / \\partial x \\, dA = \\int_c^d \\int_a^b \\partial P / \\partial x \\, dx \\, dy$. The fundamental theorem of calculus turns the inner integral into $P(b, y) - P(a, y)$. The $Q$ term is similar.\nThis is the simplest case of the [[divergence theorem|0JX86PSSA9]]. In the plane, the divergence theorem is equivalent to [[Green's theorem|0HR3N02317]].\n\n## Explanations\n\n### L1 human\n\nThe total source strength inside a rectangle equals the net flow out through its four sides.\nIn each coordinate direction, this is the fundamental theorem of calculus, applied along each line through the rectangle.\n",
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