Nonzero Jacobian Determinants of Differentiable Functions from Rn to Rn
Recall from The Jacobian Determinant of Differentiable Functions from Rn to Rn page that if $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^n$ with $\mathbf{f} = (f_1, f_2, ..., f_n)$ then the Jacobian determinant of $\mathbf{f}$ is the determinant of the $n \times n$ Jacobian matrix of $\mathbf{f}$ at $\mathbf{x}$ and is denoted $\displaystyle{\mathbf{J}_{\mathbf{f}} (\mathbf{x}) = \frac{\partial (f_1, f_2, ..., f_n)}{\partial (x_1, x_2, ..., x_n)} = \mathrm{det} (\mathbf{D} \mathbf{f} (\mathbf{c}))}$.
In linear algebra, we take considerable note of matrices with nonzero determinant. The same will be for functions whose Jacobian determinant is nonzero. We begin by looking some properties of functions with nonzero Jacobian determinants.
Theorem 1: Consider the open ball centered at $\mathbf{a}$ with radius $r > 0$, $B(\mathbf{a}, r)$, its boundary, $\partial B(\mathbf{a}, r)$, and its closure, $\overline{B(\mathbf{a}, r)} = B(\mathbf{a}, r) \cup \partial B(\mathbf{a}, r)$. Let $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}^n$ with $\mathbf{f} = (f_1, f_2, ..., f_n)$ be continuous on $\overline{B(\mathbf{a}, r)}$ and suppose that all partial derivatives in the Jacobian of $\mathbf{f}$ exist on $B(\mathbf{a}, r)$, $\mathbf{f} (\mathbf{x}) \neq \mathbf{f}(\mathbf{a})$. If $\mathbf{f} (\mathbf{x}) \neq \mathbf{f}(\mathbf{a})$ for all $\mathbf{x} \in \partial B(\mathbf{a}, r)$ and $\mathbf{J}_{\mathbf{f}}(\mathbf{x}) \neq 0$ for all $\mathbf{x} \in B(\mathbf{a}, r)$. Then the range, $\mathbf{f}(B(\mathbf{a}, r))$ contains an open ball centered at $\mathbf{f}(\mathbf{a})$. |
Theorem 2: Let $A \subseteq \mathbb{R}^n$ be open and let $\mathbf{f} : A \to \mathbb{R}^n$. If $\mathbf{f}$ is continuous and the partial derivatives of the Jacobian of $\mathbf{f}$ exist and are finite on $A$. If $\mathbf{f}$ is injective on $A$ and if $\mathbf{J}_{\mathbf{f}} (\mathbf{x}) \neq 0$ for all $\mathbf{x} \in A$ then $\mathbf{f}(A)$ is open. |
Theorem 3: Let $A \subseteq \mathbb{R}^n$ be open and let $\mathbf{f} : S \to \mathbb{R}^n$. If the partial derivatives of the Jacobian matrix of $\mathbf{f}$ are continuous on $A$ and if there exists a point $\mathbf{a} \in S$ for which $\mathbf{J}_{\mathbf{f}}(\mathbf{a}) \neq 0$ then there exists a ball centered at $\mathbf{a}$ for which $\mathbf{f}$ is injective on that ball. |
Theorem 4: Let $A \subseteq \mathbb{R}^n$ be open and let $\mathbf{f} : S \to \mathbb{R}^n$. If the partial derivatives of the Jacobian matrix of $\mathbf{f}$ are continuous on $A$ and if $\mathbf{J}_{\mathbf{f}}(\mathbf{x}) \neq 0$ for all $\mathbf{x} \in A$ then $\mathbf{f}$ is an open map. |
Recall that a mapping is said to be open if the image of every open set in the domain is open in the range.