For m functions in n variables, f_i(x₁,x₂,…,x_n) where 1 ≤ i ≤ m, the Jacobian matrix is the m×n matrix whose element in the i‐th row and j‐th column is the partial derivative $\frac{\partial f_i}{\partial x_j}.$

This Jacobian is denoted $\frac{\partial \left(f_1,f_2,\dots ,f_m\right)}{\partial \left(x_1,x_2,,\dots ,x_n\right)}.$ Jacobians satisfy a chain rule, namely:

when each x_i is a function of u₁,u₂,…,u_k. See differential (multivariate), jacobian.