Ravi Vakil's notes 13.2.D asks: "Show that if the Jacobian matrix for X = Spec $k[x_1,\ldots,x_n]/(f_1,\ldots f_r)$ has corank d at all closed points, then it has corank d at all points. (Hint: the locus where the Jacobian matrix has corank d can be described in terms of vanishing and nonvanishing of certain explicit matrices.)"
But I can't even see how to define the Jacobian matrix for non-closed points. He defines the Jacobian matrix as the $n$ by $r$ matrix such that $a_{ij} = $ the partial derivative of $f_j$ with respect to $x_i.$ If p is a closed point, then this makes sense; this partial derivative can be evaluated at p. But I don't see how he is defining the Jacobian matrix for all points, not just closed points.