Define the Jacobian $J[\mathbf{u}]$ of a smooth mapping $\mathbf{u}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$. Show that if $\mathbf{V}$ is the vector field with components

$$V_{i}=\frac{1}{3 !} \epsilon_{i j k} \epsilon_{a b c} \frac{\partial u_{a}}{\partial x_{j}} \frac{\partial u_{b}}{\partial x_{k}} u_{c}$$

then $J[\mathbf{u}]=\nabla \cdot \mathbf{V}$. If $\mathbf{v}$ is another such mapping, state the chain rule formula for the derivative of the composition $\mathbf{w}(\mathbf{x})=\mathbf{u}(\mathbf{v}(\mathbf{x}))$, and hence give $J[\mathbf{w}]$ in terms of $J[\mathbf{u}]$ and $J[\mathbf{v}]$.

Let $\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ be a smooth vector field. Let there be given, for each $t \in \mathbb{R}$, a smooth mapping $\mathbf{u}_{t}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ such that $\mathbf{u}_{t}(\mathbf{x})=\mathbf{x}+t \mathbf{F}(\mathbf{x})+o(t)$ as $t \rightarrow 0$. Show that

$$J\left[\mathbf{u}_{t}\right]=1+t Q(x)+o(t)$$

for some $Q(x)$, and express $Q$ in terms of $\mathbf{F}$. Assuming now that $\mathbf{u}_{t+s}(\mathbf{x})=\mathbf{u}_{t}\left(\mathbf{u}_{s}(\mathbf{x})\right)$, deduce that if $\nabla \cdot \mathbf{F}=0$ then $J\left[\mathbf{u}_{t}\right]=1$ for all $t \in \mathbb{R}$. What geometric property of the mapping $\mathbf{x} \mapsto \mathbf{u}_{t}(\mathbf{x})$ does this correspond to?