Consider the concept of local linearity. When looking at a nonlinear function, it will begin to look linear once zoomed in enough. # Matrix For a nonlinear matrix transformation, the Jacobian matrix represents the equivalent linear transformation observed when zoomed in and local linearity is present. In 2D: $ \mathbf{J}=\begin{bmatrix}\frac{\partial f_1}{\partial x}&\frac{\partial f_1}{\partial y}\\\frac{\partial f_2}{\partial x}&\frac{\partial f_2}{\partial y}\end{bmatrix} $ # Determinant $ |\mathbf{J}|=\begin{vmatrix}\frac{\partial f_1}{\partial x}&\frac{\partial f_1}{\partial y}\\\frac{\partial f_2}{\partial x}&\frac{\partial f_2}{\partial y}\end{vmatrix}=\frac{\partial f_1}{\partial x}\frac{\partial f_2}{\partial y}-\frac{\partial f_1}{\partial y}\frac{\partial f_2}{\partial x} $