Jacobian Matrix

Definition: Jacobian Matrix

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ be a partially differentiable real vector function with component functions $f_1,\cdots ,f_n$.

The Jacobian matrix of $f$ is the $n\times n$-matrix whose rows are the gradients of $f_1,\cdots ,f_n$:

$$\left[\begin{array}{ccc}-& (\nabla f_1{)}^{\mathsf{T}}& -\\ -& ⋮& -\\ -& (\nabla f_n{)}^{\mathsf{T}}& -\end{array}\right]$$

NOTATION

$$Df{J}_f{\mathbf{J}}_f$$

The coefficients of the Jacobian matrix are real numbers and are different for different $\mathbf{x}\in \mathcal{D}$, since the partial derivatives of $f$ depend on $\mathbf{x}$. To make this clear, the Jacobian matrix at a particular ${\mathbf{x}}_0\in \mathcal{D}$ is denoted as $Df({\mathbf{x}}_0)$.