Consider the following vector field

F$($x$,$ y$,$ z$)=$ Mi $+$ Nj $+$ Pk$.$

F$($x$,$ y$,$ z$)=$ y$2$zi $+2$xyzj $+$ xy$2$k

$($a$)$

Show that F is conservative.

curl F $=$

$($b$)$

Verify that the value of

$$C

F $$ dr

is the same for each parametric representation of C$.$

$($i$)$

C$1$: r$1($t$)=$ ti $+3$tj $+2$tk$,0$ t $1$

$$C$1$

F $$ dr $=$

$($ii$)$

C$2$: r$2()=$ sin$()$i $+3$ sin$()$j $+2$ sin$()$k$,0$

$2$

$$

$$C$2$

F $$ dr $=$