Editing Jacobian matrix and determinant (section)

== Inverse ==

According to the [[inverse function theorem]], the [[Invertible matrix|matrix inverse]] of the Jacobian matrix of an [[invertible function]] {{math|'''f''' : '''R'''<sup>''n''</sup> → '''R'''<sup>''n''</sup>}} is the Jacobian matrix of the ''inverse'' function. That is, the Jacobian matrix of the inverse function at a point {{math|'''p'''}} is

<math display="block">\mathbf J_{\mathbf{f}^{-1}}(\mathbf{p}) = {\mathbf J^{-1}_{\mathbf{f}}(\mathbf{f}^{-1}(\mathbf{p}))},</math>

and the Jacobian determinant is

<math display="block">\det(\mathbf{J}_{\mathbf{f}^{-1}}(\mathbf{p})) = \frac{1}{\det(\mathbf{J}_{\mathbf{f}}(\mathbf{f}^{-1}(\mathbf{p})))}.</math>

If the Jacobian is continuous and nonsingular at the point {{math|'''p'''}} in {{math|'''R'''<sup>''n''</sup>}}, then {{math|'''f'''}} is invertible when restricted to some [[Neighbourhood (mathematics)|neighbourhood]] of {{math|'''p'''}}. In other words, if the Jacobian determinant is not zero at a point, then the function is ''locally invertible'' near this point.

The (unproved) [[Jacobian conjecture]] is related to global invertibility in the case of a polynomial function, that is a function defined by ''n'' [[polynomial]]s in ''n'' variables. It asserts that, if the Jacobian determinant is a non-zero constant (or, equivalently, that it does not have any complex zero), then the function is invertible and its inverse is a polynomial function.