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  "body": "\n## Statement\n\nLet $V \\subseteq \\mathbb{R}^3$ be compact with a piecewise smooth boundary surface $S = \\partial V$, oriented by the outward unit normal $n$.\nIf $F$ is a vector field of class $C^1$ on an open set that contains $V$, then\n$$\\iiint_V \\nabla \\cdot F \\, dV = \\iint_S F \\cdot n \\, dS.$$\n\n## Remarks\n\nThe left side is the [[triple integral|0GTBGMZXV8]] of the [[divergence|02NSMQVQWK]], the total source strength in $V$. The right side is the outward [[flux|0YPRY7T4PK]] through the boundary.\nExample: for $F(x, y, z) = (x, y, z)$ and the unit ball, $\\nabla \\cdot F = 3$. So both sides equal $3 \\cdot \\tfrac{4}{3} \\pi = 4\\pi$.\nWith $g F$ in place of $F$, for a $C^1$ function $g$, one gets $\\iiint_V (F \\cdot \\nabla g + g \\, \\nabla \\cdot F) \\, dV = \\iint_S g \\, F \\cdot n \\, dS$. This is a form of integration by parts.\nThe same statement holds in $\\mathbb{R}^n$. For a [[rectangle in the plane|0GZ3D8F921]], the proof is a direct use of the fundamental theorem of calculus.\n\n## History\n\nLagrange discovered the theorem in 1762, and Gauss proved special cases from 1813.\nOstrogradsky gave the first proof of the general theorem in 1826, in his work on heat flow.\n\n## Explanations\n\n### L1 human\n\nWhat flows out of a closed surface must come from sources inside it.\nThe divergence theorem makes this exact: the total divergence inside equals the net flux out through the boundary.\n",
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