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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^3$ be open, let $\\phi, \\psi \\colon U \\to \\mathbb{R}$ be scalar fields, and let $A \\colon U \\to \\mathbb{R}^3$ be a vector field.\nIf these fields are of class $C^1$, then\n$$\\nabla (\\phi \\psi) = \\phi \\nabla \\psi + \\psi \\nabla \\phi, \\qquad \\nabla \\cdot (\\psi A) = \\psi \\, \\nabla \\cdot A + \\nabla \\psi \\cdot A, \\qquad \\nabla \\times (\\psi A) = \\psi \\, \\nabla \\times A + \\nabla \\psi \\times A.$$\nIf these fields are of class $C^2$, then\n$$\\nabla \\times (\\nabla \\phi) = 0, \\qquad \\nabla \\cdot (\\nabla \\times A) = 0, \\qquad \\nabla \\times (\\nabla \\times A) = \\nabla (\\nabla \\cdot A) - \\Delta A,$$\nwhere $\\Delta A$ applies the Laplacian to each component of $A$.\n\n## Remarks\n\nHere $\\nabla \\phi$ is the [[gradient|0NS2QYVC4Y]], $\\nabla \\cdot A$ the [[divergence|02NSMQVQWK]], $\\nabla \\times A$ the [[curl|0GDNHS9J2X]], and $\\Delta$ the [[Laplacian|0XQ2ZVN1PZ]].\nThe product rules follow from the product rule for partial derivatives, applied to each component.\nThe identities [[curl of a gradient is zero|0M2JRW1J0N]] and [[divergence of a curl is zero|0F56CMMC5P]] follow from the symmetry of second derivatives.\nAlso, $\\nabla \\cdot (\\nabla \\phi) = \\Delta \\phi$ and $\\Delta (\\phi \\psi) = \\phi \\Delta \\psi + 2 \\nabla \\phi \\cdot \\nabla \\psi + \\psi \\Delta \\phi$.\n\n## Explanations\n\n### L1 human\n\nThese identities are the algebra rules of vector calculus.\nThey let you expand derivatives of products and simplify repeated derivatives, the way the ordinary product rule does in one variable.\n",
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