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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $F = (F_1, \\dots, F_n) \\colon U \\to \\mathbb{R}^n$ be a [[vector field|0N67NBKQ5T]] whose [[partial derivatives|05AXG3QRKR]] $\\partial F_i / \\partial x_i$ exist on $U$.\nThe divergence of $F$ is the function $\\operatorname{div} F \\colon U \\to \\mathbb{R}$,\n$$\\operatorname{div} F = \\nabla \\cdot F = \\frac{\\partial F_1}{\\partial x_1} + \\frac{\\partial F_2}{\\partial x_2} + \\dots + \\frac{\\partial F_n}{\\partial x_n}.$$\n\n## Remarks\n\nThe divergence is the trace of the [[Jacobian matrix|0YF0VXKXGB]] of $F$.\nFor $F(x) = x$ on $\\mathbb{R}^n$, $\\operatorname{div} F = n$. For the rotation field $F(x, y) = (-y, x)$, $\\operatorname{div} F = 0$.\nThe divergence is linear, and $\\operatorname{div}(\\phi F) = \\nabla \\phi \\cdot F + \\phi \\operatorname{div} F$ for a differentiable function $\\phi$.\nTake a $C^1$ field and a small region around $x_0$. The divergence at $x_0$ is the limit of the outward flux through the boundary of the region, divided by its volume. This follows from the [[divergence theorem|0JX86PSSA9]].\n\n## Explanations\n\n### L1 human\n\nThink of $F$ as the velocity of a fluid.\nWhere the divergence is positive, more fluid leaves a small region than enters it, as at a source. Where it is negative, fluid gathers, as at a sink.\n",
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