B1. Jacobian Matrix

The Jacobian matrix of a vector function $\vec{f}(\vec{x})= (f_{1}(\vec{x}), \dots ,f_{m}(\vec{x}))^{\cal T}$ ( $\vec{x} \in \mathbb{R}^{n}$) is defined by the matrix $\mathcal{J} \in \mathbb{R}^{m \times n}$

$$\mathcal{J}(\vec{x}) = \mathop{\rm Jac}\nolimits (\vec{f}(\... ...tial f_{m}(\vec{x})}{\partial x_n} \\ \end{array} \right ) .$$ (B1)

The Jacobian can also be written by the transposed gradient vector $f_{i}(\vec{x})$ for each row

$$\mathop{\rm Jac}\nolimits (\vec{x}) = \left ( \begin{ar... ...mits f_{m}(\vec{x}) \right )^{\cal T} \end{array} \right ) .$$ (B2)