Jacobian Matrix

Definition

Let $f : R^{n} \to R^{m}$ be a vector function with components $f_{1}, f_{2}, \dots, f_{m}$ .

The Jacobian matrix of $f$ with respect to $x = [x_{1} \dots x_{n}]^{T}$ is:

J_{f} (x) = \frac{\partial f}{\partial x ^{T}} = \frac{\partial f _{1}}{\partial x _{1}} \frac{\partial f _{2}}{\partial x _{1}} ⋮ \frac{\partial f _{m}}{\partial x _{1}} \frac{\partial f _{1}}{\partial x _{2}} \frac{\partial f _{2}}{\partial x _{2}} ⋮ \frac{\partial f _{m}}{\partial x _{2}} \dots \dots ⋱ \dots \frac{\partial f _{1}}{\partial x _{n}} \frac{\partial f _{2}}{\partial x _{n}} ⋮ \frac{\partial f _{m}}{\partial x _{n}}

This is an $m \times n$ matrix where the $i$ -th row is $\nabla f_{i}^{T}$ .

Let $f : R^{2} \to R^{3}$ be:

f ([x_{1} x_{2}]) = x_{1}^{2} + x_{2} 2 x_{1} x_{2} x_{2}^{2}

J_{f} = \frac{\partial f _{1}}{\partial x _{1}} \frac{\partial f _{2}}{\partial x _{1}} \frac{\partial f _{3}}{\partial x _{1}} \frac{\partial f _{1}}{\partial x _{2}} \frac{\partial f _{2}}{\partial x _{2}} \frac{\partial f _{3}}{\partial x _{2}} = 2 x_{1} 2 x_{2} 0 1 2 x_{1} 2 x_{2}

At $x = [12]$ :

J_{f} ([12]) = 240124