Jacobian Matrix

Topics: Differentiable Vectorial Function - Matrix

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(definition)

In the context of partial derivatives, the Jacobian matrix for a function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is defined as:

$$J(x_1,x_2,\dots ,x_n)=\left(\begin{array}{cccc}\frac{\partial f_1}{\partial x_1}& \frac{\partial f_1}{\partial x_2}& \dots & \frac{\partial f_1}{\partial x_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial f_m}{\partial x_1}& \frac{\partial f_m}{\partial x_2}& \dots & \frac{\partial f_m}{\partial x_n}\end{array}\right)$$

…where $f=(f_1,f_2,\dots ,f_n)$.

Example

Let $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be defined by $f(x,y)=(2x+3y,-\cos (xy))$.

We got:

$$f_1={\mathbb{R}}^2\to \mathbb{R},f_1(x,y)=2x+3y$$

$$f_2={\mathbb{R}}^2\to \mathbb{R},f_2(x,y)=-\cos (xy)$$

So our Jacobian matrix is:

$$J(x,y)=\left(\begin{array}{cc}\frac{\partial f_1}{\partial x}& \frac{\partial f_1}{\partial y}\\ \frac{\partial f_2}{\partial x}& \frac{\partial f_2}{\partial y}\end{array}\right)$$

$$J(x,y)=\left(\begin{array}{cc}2& 3\\ y\sin (xy)& x\sin (xy)\end{array}\right)$$