The Jacobian Determinant of Differentiable Functions from Rn to Rn

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}