$1\mathrm{.}31.$ Let $u(x,y,z,t)be$ a mass density defined $in$ all $of3D$ space, governed $by$ the advec$-$

tion flux rule with a nonzero velocity vector vec$(v).$ Assume there are $no$ sources. Let $Cbea$

unit$-$length cube test volume, and let $Ebe$ another connected and bounded test volume

that contains all points around $C$ but does not include any points $ofC.$ Suppose $at$ time

$t$ that every point $inC$ possesses positive mass density, and there $is$ zero mass elsewhere

$in$ all $3D$ space, including $E.$ Determine the following true, false, $or$ indeterminate and

explain your answer:

$(a)$ The mass inside $C$ will $be$ declining $at$ time $t.$

$(b)$ The net flux out $ofC$ will $be$ positive $at$ time $t.$

$(c)$ The mass inside $E$ will $be$ decreasing $at$ time $t.$

$(d)$ The flux out $ofC$ will $limitto$ zero $ast.$