As a function of time, the electric current flowing out of a volume $$ surrounded by closed

surface $$ is given by

$()=(,)$

$$

$$

where $(,)$ is the current density in $$ at time $,$ and $$ points out of $.$ Also, if $$ is the total

charge in $,$ then

$()=$

$$

$$

since a charge $$ that flows out of $$ in time $$ depletes $$ inside $$ by $($assuming charge is

conserved$).$

$($a$)$ Show that the above equations imply

$(,)=$

$(,)$

$$