Vector Calculus | Part IA, 2002

Prove that

×(a×b)=abba+(b)a(a)b\nabla \times(\mathbf{a} \times \mathbf{b})=\mathbf{a} \nabla \cdot \mathbf{b}-\mathbf{b} \nabla \cdot \mathbf{a}+(\mathbf{b} \cdot \nabla) \mathbf{a}-(\mathbf{a} \cdot \nabla) \mathbf{b}

SS is an open orientable surface in R3\mathbb{R}^{3} with unit normal n\mathbf{n}, and v(x)\mathbf{v}(\mathbf{x}) is any continuously differentiable vector field such that nv=0\mathbf{n} \cdot \mathbf{v}=0 on SS. Let m\mathbf{m} be a continuously differentiable unit vector field which coincides with n\mathbf{n} on SS. By applying Stokes' theorem to m×v\mathbf{m} \times \mathbf{v}, show that

S(δijninj)vixjdS=Cuvds\int_{S}\left(\delta_{i j}-n_{i} n_{j}\right) \frac{\partial v_{i}}{\partial x_{j}} d S=\oint_{C} \mathbf{u} \cdot \mathbf{v} d s

where ss denotes arc-length along the boundary CC of SS, and u\mathbf{u} is such that uds=ds×n\mathbf{u} d s=d \mathbf{s} \times \mathbf{n}. Verify this result by taking v=r\mathbf{v}=\mathbf{r}, and SS to be the disc rR|\mathbf{r}| \leqslant R in the z=0z=0 plane.

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