Non-singular point (definition) (reviewed)

Let $V$ be a variety over a perfect field $F$. A point $P$ of $V$ is non-singular if the module of differentials of $V$ is locally free at $P$. According to the Jacobian criterion, if $V$ is defined in a neighborhood of $P$ by affine polynomial equations $f_1(X_1, \ldots, X_n) = \ldots =f_r(X_1, \ldots, X_n)=0$, then $V$ is non-singular at $P$ if the Jacobian matrix $\left( \frac{\partial f_i}{\partial X_j} \right)_{ij}$ has the same rank as the codimension of $V$ in $\mathbb A^n$.