Let $k$ be a field of characteristic $0$. Let $f_1, \dots, f_m \in k[x_1, \dots, x_n]$.
Are $f_1, \dots , f_m$ algebraically independent over $k$ if and only if the rank of the Jacobian matrix $(\frac{\partial f_i}{\partial x_j})_{m \times n}$ is $m$?
Note that the "if" part is easy.
A example for "only if" part.
Let $f_1 = x_1, f_2 = x_2x_3, f_3 = x_1x_2x_3$. The Jacobian matrix is $\left( \begin{array}{ccc} 1 & 0 & 0\\ 0 & x_3 & x_2\\ x_2x_3 & x_1x_3 & x_1x_2\\ \end{array} \right)$. The rank is 2. They are algebraically dependent since $f_1f_2 - f_3 = 0$.