Jacobian Matrix

Definition

Suppose $\mathbf{f}:{\mathbb{R}}^n\to {\mathbb{R}}^m,\mathbf{f}(\mathbf{x})=(f_1(\mathbf{x}),\dots ,f_m(\mathbf{x}))$ where $\mathbf{x}\in {\mathbb{R}}^n$. Then, the $m\times n$ Jacobian Matrix of $\mathbf{f}$ is defined as ${\mathbf{J}}_{m\times n}=\left(\begin{array}{ccc}\frac{\partial \mathbf{f}}{\partial x_1}& \dots & \frac{\partial \mathbf{f}}{\partial x_n}\end{array}\right)=\left(\begin{array}{c}{\nabla }^{⊺}{f}_1(p)\\ ⋮\\ {\nabla }^{⊺}{f}_m(p)\end{array}\right)=\left(\begin{array}{ccc}\frac{\partial f_1}{\partial x_1}& \dots & \frac{\partial f_1}{\partial x_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial f_m}{\partial x_1}& \dots & \frac{\partial f_m}{\partial x_n}\end{array}\right)$ where ${\nabla }^{⊺}{f}_i$ is the transpose of the Gradient of the $i$-th component of the function.