Equation rows and coordinate columns in an affine Jacobian

Let $k$ be a field, let $I\subseteq k[t_1,\dots ,t_n]$ be an ideal with a specified finite generating list $f_1,\dots ,f_r$, and let $a=(a_1,\dots ,a_n)\in k^n$ satisfy $f(a)=0$ for every $f\in I$. The Jacobian matrix at $a$, with the equation-row convention, is the $r\times n$ matrix $J_{(f_1,\dots ,f_r)}(a)={(\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{t}_j}(a))}_{1\le i\le r,1\le j\le n}\mathrm{.}$ Thus row $i$ is the differential of equation $f_i$, and column $j$ corresponds to coordinate $t_j$. Formal derivatives are computed on monomials by ${\mathrm{\partial }}_{t_j}(t_1^{e_1}\cdots t_n^{e_n})=\{\begin{array}{ll}{e}_j{t}_1^{e_1}\cdots t_j^{e_j-1}\cdots t_n^{e_n},& e_j>0,\\ 0,& e_j=0.\end{array}$ with the integer coefficient read in $k$, and are extended $k$-linearly. The definition uses the actual scheme ideal $I$; it does not assume that $I$ is radical or that $k$ is perfect. For a reduced classical algebraic set over an algebraically closed field, this specializes to its coordinate ring $k[X]=k[t_1,\dots ,t_n]\mathrm{/}I(X)$ (The coordinate ring of an affine algebraic set).

For a scheme-theoretic affine zero locus, an equation list for the same underlying point set is not substituted for the actual ideal: nilpotent structure changes the Jacobian problem.