State the divergence theorem for a vector field $\mathbf{u}(\mathbf{x})$ in a region $V$ of $\mathbb{R}^{3}$ bounded by a smooth surface $S$.

Let $f(x, y, z)$ be a homogeneous function of degree $n$, that is, $f(k x, k y, k z)=$ $k^{n} f(x, y, z)$ for any real number $k$. By differentiating with respect to $k$, show that

$$\mathbf{x} \cdot \nabla f=n f$$

Deduce that

$$\int_{V} f \mathrm{~d} V=\frac{1}{n+3} \int_{S} f \mathbf{x} \cdot \mathbf{d} \mathbf{A}$$

Let $V$ be the cone $0 \leqslant z \leqslant \alpha, \alpha \sqrt{x^{2}+y^{2}} \leqslant z$, where $\alpha$ is a positive constant. Verify that $(\dagger)$ holds for the case $f=z^{4}+\alpha^{4}\left(x^{2}+y^{2}\right)^{2}$.