(Solved): 1. (a) Let \( \mathbf{F}=F_{1}(x, y, z) \mathbf{i}+F_{2}(x, y, z) \mathbf{j}+F_{3}(x, y, z) \mathbf ...

1. (a) Let \( \mathbf{F}=F_{1}(x, y, z) \mathbf{i}+F_{2}(x, y, z) \mathbf{j}+F_{3}(x, y, z) \mathbf{k} \) denote a smooth vector field on \( \mathbb{R}^{3} \). Give a definition of the circulation density of \( \mathbf{F} \) and the divergence of \( \mathbf{F} \). \( [4] \) (b) Let \( f \) and \( g: \mathbb{R}^{3} \rightarrow \mathbb{R} \) denote two smooth scalar fields, and let \( S \subset \mathbb{R}^{3} \) denote a nonempty bounded surface equipped with a continuous unit normal vector field \( \mathbf{n} \), and with boundary curve C. Assuming the hypotheses of Stokes' Theorem are met, and proving any necessary vector identities, show that \[ \iint_{S}(\nabla(f) \times \nabla(g)) \cdot \mathbf{n d} S=\oint_{C} f \nabla(g) \cdot \mathrm{d} \mathbf{r}, \] for some parameterisation \( \mathbf{r} \) of \( C \). [7] (c) Let \( D \) denote a non empty bounded open subset of \( \mathbb{R}^{3} \), let \( S=\partial D \) denote the boundary of \( D \) and let \( \mathbf{n} \) denote a continuous unit normal vector field to \( S \), if it exists. Under the assumption that the hypotheses of Gauss' Divergence Theorem are met, prove that the volume \( V \) of \( D \) satisfies the following identity. \[ V=\frac{1}{3} \iint_{S} \mathbf{r} \cdot \mathbf{n d S} \] Here \( \mathbf{r} \) denotes the position vector of the point \( (x, y, z) \). [6]