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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}^n$, and let $a \\in U$ be a point at which all first partial derivatives of $f$ exist.\nThe Jacobian determinant of $f$ at $a$ is the determinant\n$$\\det J_f(a) = \\det \\left( \\frac{\\partial f_i}{\\partial x_j}(a) \\right)_{1 \\le i, j \\le n}$$\nof the [[Jacobian matrix|0YF0VXKXGB]] of $f$ at $a$.\nIt is also written $\\dfrac{\\partial (f_1, \\dots, f_n)}{\\partial (x_1, \\dots, x_n)}(a)$, and it is often called the Jacobian of $f$.\n\n## Remarks\n\nFor the polar map $(r, \\theta) \\mapsto (r \\cos \\theta, r \\sin \\theta)$, the Jacobian determinant is $r \\cos^2 \\theta + r \\sin^2 \\theta = r$.\nIf $f$ is of class $C^1$ and $\\det J_f(a) \\ne 0$, then $f$ has a differentiable local inverse near $a$, by the [[inverse function theorem|0K9HSJFCMJ]].\nThe absolute value $|\\det J_f|$ is the factor in the [[change of variables formula|00N5FHP1CV]] for multiple integrals.\n\n## Explanations\n\n### L1 human\n\nA linear map of the plane sends the unit square to a parallelogram, and the determinant is the signed area of that parallelogram.\nThe Jacobian determinant does the same for a curved map, at the scale of tiny squares near one point.\n",
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