Editing Jacobian matrix and determinant (section)

== Critical points ==

{{main|Critical point (mathematics)|l1=Critical point}}

If {{math|'''f''' : '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>}} is a [[differentiable function]], a ''critical point'' of {{math|'''f'''}} is a point where the [[rank (linear algebra)|rank]] of the Jacobian matrix is not maximal. This means that the rank at the critical point is lower than the rank at some neighbour point. In other words, let {{math|''k''}} be the maximal dimension of the [[open ball]]s contained in the image of {{math|'''f'''}}; then a point is critical if all [[minor (linear algebra)|minor]]s of rank {{math|''k''}} of {{math|'''f'''}} are zero.

In the case where {{math|1=''m'' = ''n'' = ''k''}}, a point is critical if the Jacobian determinant is zero.