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In multivariable calculus, the jacobian matrix of a smooth map at a given point is the matrix of its partial derivatives evaluated at this point.

Let $f\colon U\rightarrow\mathbb{R}^n$ be a map on an open subset $U$ of $\mathbb{R}^m$ and let $x\in U$.

Definition 1. Assume that $f$ is differentiable at $x$, then the jacobian matrix of $f$ at $x$, denoted by $\textrm{Jac}_xf$ is the matrix of the linear map $\mathrm{d}_xf\colon\mathbb{R}^m\rightarrow\mathbb{R}^n$ in the canonical basis of $\mathbb{R}^m$ and $\mathbb{R}^n$.

One has the following:

Proposition 1. With the same assumption than in definition 1, if $f:=(f_1,\cdots,f_n)$, then one has: $$\textrm{Jac}_xf=\left(\frac{\partial f_j}{\partial x_i}(x)\right)_{1\leqslant i\leqslant m,1\leqslant j\leqslant n}.$$

Remark 1. If $n=1$, then the jacobian matrix of $f$ at $x$ is the gradient of $f$ at $x$, namely one has: $$\textrm{Jac}_xf=\nabla_xf.$$

Definition 2. With the same assumption than in definition 1, the jacobian of $f$ at $x$ is the determinant of the jacobian matrix of $f$ at $x$, $\textrm{Jac}_xf$.