Jacobian ideal

In mathematics the Jacobian ideal or gradient ideal is the ideal generated by the Jacobian of a function or function germ. Let ${\mathcal {O}}(x_{1},\ldots ,x_{n})$ denote the ring of smooth functions in $n$ variables and $f$ a function in the ring. The Jacobian ideal of $f$ is

J_{f}:=\left\langle {\frac {\partial f}{\partial x_{1}}},\ldots ,{\frac {\partial f}{\partial x_{n}}}\right\rangle .

Jacobian ideal

See also