ome $>$ Math $254-01$ Summer $2025>$ Assessment

Homework $6\mathrm{.}8$ The Divergence Theorem

Score: $6/15$ Answered: $4/9$

Question $5$

Score on last try: $0$ of $2$ pts$.$ See Details for more.

$$ You can retry this question below

Let vec$(v)=($:$2,ysinz,cosz$:$)$ be the velocity field of a fluid. Compute the flux of vec$(v)$ across the surface $(x-8{)}^2=4y^{2}16z^2$ where $q,0$ and the surface $is$ oriented away from the origin.

Hint: This $is$ a very tricky question. $If$ the total divergence $is$ zero, that does not necessarily mean there $isno$ flow through the closed surface formed $by$ the cone and its ellipse$-$shaped base. $It$ mans the flow into the closed surface $is$ equal $to$ the flow out $of$ the clased surface. Assuming the divergence $is$ zero $(you$ should checkl$),$ the flow $(or$ flux$)in$ through the ellipse$-$shaped base will equal the flow $(or$ flux$)$ out $of$ the cone. Thun, find a way $to$ calculate the flow throuph the ellipse$-$shaped base $of$ the closed surface. $7$Let vec$(v)=($:$2,ysinz,cosz$:$)$ be the velocity field of a fluid. Compute the flux of vec$(v)$ across the surface $(x-8{)}^2=4y^{2}16z^2$ where $0$ and the surface $is$ oriented away from the origin.

Hint: This $is$ a very tricky question. $If$ the total divergence $is$ zero, that does not necessarily mean there $isno$ flow through the closed surface formed $by$ the cone and its ellipse$-$shaped base. $It$ means the flow into the closed surface $is$ equal $to$ the flow out $of$ the closed surface. Assuming the divergence $is$ zero $(you$ should check!$),$ the flow $(or$ flux$)in$ through the ellipse$-$shaped base will equal the flow $(or$ flux$)$ out $of$ the cone. Thus, find a way $to$ calculate the flow through the ellipse$-$shaped base $of$ the closed surface. $$