Jacobian | Sahithyan's Notes

The Jacobian generalizes the derivative of a scalar function to a vector-valued function. 2 objects carry the name.

The Jacobian matrix
The matrix of first partial derivatives of a vector-valued function.
The Jacobian determinant
The determinant of the Jacobian matrix of a transformation.

Jacobian Matrix

For $f : \mathbb{R}^n \to \mathbb{R}^m$ with coordinate functions $f_1, f_2, \ldots, f_m$ , the Jacobian matrix $\mathbf{J}_p f(\mathbf{x})$ is the $m \times n$ matrix of first partial derivatives

\mathbf{J}_p f(\mathbf{x}) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{pmatrix}

Row $i$ of $\mathbf{J}_p f(\mathbf{x})$ is the gradient of $f_i$ .

Jacobian Determinant

For a transformation $T$ , the Jacobian determinant is the determinant of the Jacobian matrix of $T$ .

For $T$ from the $uv$ -plane to the $xy$ -plane, $x = x(u,v)$ , $y = y(u,v)$ :

J(u,v) = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\[6pt] \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \frac{\partial x}{\partial u}\frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial y}{\partial u}

For $T$ from $uvw$ -space to $xyz$ -space, $x = x(u,v,w)$ , $y = y(u,v,w)$ , $z = z(u,v,w)$ :

J(u,v,w) = \frac{\partial(x,y,z)}{\partial(u,v,w)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\[6pt] \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\[6pt] \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{vmatrix}

Examples:

$\dfrac{\partial(x,y)}{\partial(r,\theta)} = r$ , for polar coordinates $x = r\cos\theta$ , $y = r\sin\theta$
$\dfrac{\partial(x,y,z)}{\partial(\rho,\phi,\theta)} = \rho^2\sin\phi$ , for spherical coordinates

First-Order Approximation

The Jacobian matrix gives the first-order approximation of $f$ around $\mathbf{x}_0$ . See Approximation.

f(\mathbf{x}) \approx f(\mathbf{x}_0) + \mathbf{J}_p f(\mathbf{x}_0)\,(\mathbf{x} - \mathbf{x}_0)

For scalar $f$ , the gradient form is

f(\mathbf{x}) \approx f(\mathbf{x}_0) + \nabla f(\mathbf{x}_0)^T(\mathbf{x} - \mathbf{x}_0)

Worked Examples

Example 1

Find the Jacobian matrix of the system

\begin{aligned} f_1(\mathbf{x}) &= 3x_1 - \cos(x_2 x_3) - \tfrac{1}{2} \\ f_2(\mathbf{x}) &= x_1^2 - 81(x_2 + 0.1)^2 + \sin x_3 + 1.06 \\ f_3(\mathbf{x}) &= e^{-x_1 x_2} + 20x_3 + \tfrac{10\pi - 3}{3} \end{aligned}

Solution Reveal Hide

The partial derivatives give

J(\mathbf{x}) = \begin{pmatrix} 3 & x_3 \sin(x_2 x_3) & x_2 \sin(x_2 x_3) \\ 2x_1 & -162(x_2 + 0.1) & \cos x_3 \\ -x_2 e^{-x_1 x_2} & -x_1 e^{-x_1 x_2} & 20 \end{pmatrix}

Example 2

Find the Jacobian matrix of $f_1 = x_1^2 + x_2^2$ and $f_2 = x_1 x_2$ , then evaluate it at $\mathbf{x} = (1, 2)^T$ .

Solution Reveal Hide

J(\mathbf{x}) = \begin{pmatrix} 2x_1 & 2x_2 \\ x_2 & x_1 \end{pmatrix}

J(1, 2) = \begin{pmatrix} 2 & 4 \\ 2 & 1 \end{pmatrix}