Answer
Result : See proof
Work Step by Step
Step 1 We know that the flux of a vector field $\vec{F}$ over a closed surface $\Sigma$ is related to the divergence of $\vec{F}$ through the Divergence Theorem: \[ \iint_{\Sigma} \vec{F} \cdot \mathbf{n} \, dS = \iiint_{G} \text{div}(\vec{F}) \, dV, \] where $\mathbf{G}$ is the volume enclosed by $\Sigma$. We are asked to find applications of equation (1). Step 2 One of the important applications of equation (1) is identifying the sources and the sinks of the vector field $\vec{F}$. If we take a certain region $G$ in $\mathbb{R}^3$ enclosed by $\Sigma$, if the number of field lines that enter $G$ is higher than the number that leaves it, there is a sink in that region. Vice versa, there is a source in $G$. The number of field lines entering/leaving $\Sigma$ is given by the flux (left side of equation (1)) of a vector field $\vec{F}$. Step 3 From equation (1), the divergence at the point $\mathbf{P}_0$ can be defined through the flux density as: \[ \text{div}(\vec{F}(\mathbf{P}_0)) = \lim_{{\text{vol}(G) \to 0}} \frac{1}{\text{vol}(G)} \iint_{\Sigma} \vec{F} \cdot \mathbf{n} \, dS. \] Thus, divergence can be used to define sinks and sources. If the divergence is positive (the flux on the right is positive), then there is a source at $\mathbf{P}_0$, if it is negative (the flux on the right is negative), there is a sink, and if it is $0$, there are no sinks and sources. The vector flow fields used in hydrodynamics that satisfy $\text{div}(\vec{F}) = 0$ are called incompressible flow fields. Step 4 Another great application of equation (1) is used for Inverse-Square Fields, meaning the fields that fall off into distance $r$ with $\frac{1}{r^2}$: \[ \vec{F}(r) = \alpha \frac{\mathbf{r}}{r^3}, \] where $\alpha$ is a constant. It can be shown for this field that, through the use of the Divergence Theorem (equation (1)), we can find the flux of $\vec{F}$ through the closed surface $\Sigma$ as: \[ \iint_{\Sigma} \vec{F} \cdot \mathbf{n} \, dS = \iint_{\Sigma} \frac{\mathbf{r}}{r^3} \cdot \mathbf{n} \, dS = 4\pi\alpha. \] This can simplify many calculations and is important due to some fundamental fields, such as electrostatic and gravitational fields, being Inverse-Square Fields.
