Understanding the Jacobian Matrix: a step by step introduction with examples

The Jacobian matrix is an important tool in multivariable calculus, used to represent the partial derivatives of a system of functions. If we have a system of functions from the space $\mathbb{R}^n$ to $\mathbb{R}^m$ , the Jacobian matrix provides a way to express how the output functions change with respect to the input variables.

Definition:

Let $\mathbf{F} : \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a vector-valued function, defined as:

$\mathbf{F}(\mathbf{x}) = \begin{pmatrix} f_1(x_1, x_2, \ldots, x_n) \\ f_2(x_1, x_2, \ldots, x_n) \\ \vdots \\ f_m(x_1, x_2, \ldots, x_n) \end{pmatrix}$

The Jacobian matrix of $\mathbf{F}$ at the point $\mathbf{x} = (x_1, x_2, \ldots, x_n)$ , denoted by $J$ , is the matrix whose entries are the partial derivatives of the functions $f_i$ with respect to the variables $x_j$ :

$J = \begin{pmatrix}\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\\vdots & \vdots & \ddots & \vdots \\\frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n}\end{pmatrix}$

Example 1:

Let’s consider a system of functions:

$\mathbf{F}(x, y) = \begin{pmatrix} f_1(x, y) \\ f_2(x, y) \end{pmatrix} = \begin{pmatrix} x^2 + y^2 \\ xy \end{pmatrix}$

The Jacobian matrix $J$ of this system is:

$J = \begin{pmatrix}\frac{\partial (x^2 + y^2)}{\partial x} & \frac{\partial (x^2 + y^2)}{\partial y} \\\frac{\partial (xy)}{\partial x} & \frac{\partial (xy)}{\partial y}\end{pmatrix} = \begin{pmatrix} 2x & 2y \\ y & x \end{pmatrix}$

Interpretation:

The Jacobian matrix helps analyze and understand how multivariable functions change, especially when studying continuity, differentiability, and other properties. It’s also very useful in optimization and solving nonlinear systems of equations.

Example 2:

Consider the function from $\mathbb{R}^3$ to $\mathbb{R}^2$ , defined as:

$\mathbf{F}(x, y, z) = \begin{pmatrix} f_1(x, y, z) \\ f_2(x, y, z) \end{pmatrix} = \begin{pmatrix} xy + z^2 \\ x + yz \end{pmatrix}$

Compute the Jacobian matrix $J$ at point $(x, y, z)$ :

$J = \begin{pmatrix}\frac{\partial (xy + z^2)}{\partial x} & \frac{\partial (xy + z^2)}{\partial y} & \frac{\partial (xy + z^2)}{\partial z} \\\frac{\partial (x + yz)}{\partial x} & \frac{\partial (x + yz)}{\partial y} & \frac{\partial (x + yz)}{\partial z}\end{pmatrix}$

Compute the partial derivatives:

$\frac{\partial (xy + z^2)}{\partial x} = y$
$\frac{\partial (xy + z^2)}{\partial y} = x$
$\frac{\partial (xy + z^2)}{\partial z} = 2z$
$\frac{\partial (x + yz)}{\partial x} = 1$
$\frac{\partial (x + yz)}{\partial y} = z$
$\frac{\partial (x + yz)}{\partial z} = y$

So the Jacobian matrix is:

$J = \begin{pmatrix} y & x & 2z \\ 1 & z & y \end{pmatrix}$

Interpretation:

This Jacobian shows how the functions $f_1$ and $f_2$ change with respect to the variables $x$ , $y$ , and $z$ . It’s particularly useful for analyzing the sensitivity of the system to input variables.