The Jacobian Determinant of Differentiable Functions from Rn to Rn

We are about to examine various properties of multivariable functions from $\mathbb{R}^n$ to $\mathbb{R}^n$. We will begin by investigating a very significant property known as the Jacobian determinant.

Definition: Let $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^n$ with $\mathbf{f} = (f_1, f_2, ..., f_n)$. The Jacobian Determinant of $\mathbf{f}$ at $\mathbf{x}$ is the determinant of the $n \times n$ Jacobian matrix $\displaystyle{\mathbf{D} \mathbf{f} (\mathbf{x}) = \begin{bmatrix} \frac{\partial f_1(\mathbf{x})}{\partial x_1} & \frac{\partial f_1(\mathbf{x})}{\partial x_2} & \cdots & \frac{\partial f_1(\mathbf{x})}{\partial x_n}\\ \frac{\partial f_2(\mathbf{x})}{\partial x_1} & \frac{\partial f_2(\mathbf{x})}{\partial x_2} & \cdots & \frac{\partial f_2(\mathbf{x})}{\partial x_n}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial f_n(\mathbf{x})}{\partial x_1} & \frac{\partial f_n(\mathbf{x})}{\partial x_2} & \cdots & \frac{\partial f_n(\mathbf{x})}{\partial x_n} \end{bmatrix}}$ and is denoted $\mathbf{J}_{\mathbf{f}}(\mathbf{x}) = \mathrm{det} (\mathbf{D} \mathbf{f} (\mathbf{c}))$ or $\displaystyle{\frac{\partial (f_1, f_2, ..., f_n)}{\partial (x_1, x_2, ..., x_n)} = \mathrm{det} (\mathbf{D} \mathbf{f} (\mathbf{c}))}$.

Note that the differentiability of $\mathbf{f}$ at $\mathbf{x}$ is sufficiently but not necessary for the existence of the Jacobian determinant of $\mathbf{f}$ at $\mathbf{x}$. All that we require to determine the Jacobian determinant of $\mathbf{f}$ at $\mathbf{x}$ is the existence of all first order partial derivatives of the component functions $f_1, f_2, ..., f_n$ at $\mathbf{x}$.

For example, consider the function $\mathbf{f} : \mathbb{R}^2 \to \mathbb{R}^2$ defined by:

(1)

\begin{align} \quad \mathbf{f}(x, y) = (x^2 + y^2, 2xy) \end{align}

Then $f_1(x, y) = x^2 + y^2$ and $f_2(x, y) = 2xy$. We see that:

(2)

\begin{align} \quad \frac{\partial f_1}{\partial x} = 2x \quad , \quad \frac{\partial f_1}{\partial y} = 2y \quad , \quad \frac{\partial f_2}{\partial x} = 2y \quad , \quad \frac{\partial f_2}{\partial y} = 2x \end{align}

Therefore the Jacobian matrix of $\mathbf{f}$ at $(x, y)$ is given by:

(3)

\begin{align} \quad \mathbf{D} \mathbf{f} (x, y) = \begin{bmatrix} 2x & 2y\\ 2y & 2x \end{bmatrix} \end{align}

And the Jacobian determinant of $\mathbf{f}$ at $(x, y)$ is given by:

(4)

\begin{align} \quad \frac{\partial (f_1, f_2)}{\partial (x, y)} = 4x^2 - 4y^2 = 4(x^2 - y^2) \end{align}