A body of volume $V$ lies totally submerged in a motionless fluid of uniform density $\rho$. Show that the force $\mathbf{F}$ on the body is given by

$$\mathbf{F}=-\int_{S}\left(p-p_{0}\right) \mathbf{n} d S$$

where $p$ is the pressure in the fluid and $p_{0}$ is atmospheric pressure. You may use without proof the generalised divergence theorem in the form

$$\int_{S} \phi \mathbf{n} d S=\int_{V} \boldsymbol{\nabla} \phi d V$$

Deduce that

$$\mathbf{F}=\rho g V \hat{\mathbf{z}},$$

where $\hat{\mathbf{z}}$ is the vertically upward unit vector. Interpret this result.