The unit sphere in $\mathbb{R}^n$, with respect to the Euclidean norm $\|\cdot\|$, is
$$B_1(0) = \{x \in \mathbb{R}^n : \|x\| \leq1 \}.$$
Is it possible to compute the Lebesgue measure of this set using the Divergence Theorem? Can you give me a hint about the vector field I should use?
Let $F(x_1, \dots, x_n)= (x_1, \dots, x_n)$. From the Divergence Theorem we have $$\int_{B_1(0)} \nabla \cdot \mathbf{F} \space dV = \int_{\partial B_1(0)} \mathbf{F} \cdot \mathbf{\nu} \space dS, $$ where $\mathbf{\nu}$ is the normal unit vector to $\partial B_1(0)$.
Hence we have $$\int_{B_1(0)} \nabla \cdot \mathbf{F} \space dV = n\int_{B_1(0)} \space dV = n \cdot \mathcal{L}^n (B_1(0)) = \int_{\partial B_1(0)} dS = \mathcal{L}^{n-1} (\partial B_1(0)),$$ where $\mathcal{L}^n$ denotes the $n$-dimensional Lebesgue measure. So we proved that the measure of the unit ball in $\mathbb{R}^n$ is equal to the the measure of its surface divided by the dimension $n$.