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  "body": "\n## Summary\n\nThe ring of differential operators $D(A)$ of a commutative algebra $A$ over a field $k$ is a ring of $k$-linear maps $A \\to A$.\nIts elements behave like linear differential operators of finite order.\nGrothendieck's definition uses only commutators with multiplication maps, so it makes sense for any commutative algebra, without coordinates.\nFor the polynomial ring $k[x_1, \\dots, x_n]$ in characteristic $0$ it is the [[Weyl algebra|98QE7KY55W]] $A_n(k)$.\nIts modules, called $D$-modules, are a tool for studying linear partial differential equations by algebra.\n\n## Explanations\n\n### L1 human\n\nA differential operator such as $f \\mapsto x^2 f'' + 3 f'$ combines multiplications by functions with derivatives.\nCommuting such an operator with a multiplication by a function lowers its order by one, by the product rule.\nAn operator of order $0$ is just a multiplication, and the definition builds the higher orders from there.\nFor polynomials in one variable over the rationals, every such operator is a finite sum $\\sum_{j} a_j(x) \\, d^j/dx^j$ with polynomial coefficients $a_j$.\n",
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