Paper 3, Section II, C

Vector Calculus | Part IA, 2016

Define the Jacobian J[u]J[\mathbf{u}] of a smooth mapping u:R3R3\mathbf{u}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}. Show that if V\mathbf{V} is the vector field with components

Vi=13!ϵijkϵabcuaxjubxkucV_{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[u]=VJ[\mathbf{u}]=\nabla \cdot \mathbf{V}. If v\mathbf{v} is another such mapping, state the chain rule formula for the derivative of the composition w(x)=u(v(x))\mathbf{w}(\mathbf{x})=\mathbf{u}(\mathbf{v}(\mathbf{x})), and hence give J[w]J[\mathbf{w}] in terms of J[u]J[\mathbf{u}] and J[v]J[\mathbf{v}].

Let F:R3R3\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} be a smooth vector field. Let there be given, for each tRt \in \mathbb{R}, a smooth mapping ut:R3R3\mathbf{u}_{t}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} such that ut(x)=x+tF(x)+o(t)\mathbf{u}_{t}(\mathbf{x})=\mathbf{x}+t \mathbf{F}(\mathbf{x})+o(t) as t0t \rightarrow 0. Show that

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

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

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