Determine whether the statement is true or false. If it is true,

explain why. If it is false, explain why or give an example that dis$-$

proves the statement.

If F is a vector field, then divF is a vector field.

If F is a vector field, then curl F is a vector field.

If f has continuous partial derivatives of all orders on R$^(3),$

then div$($curlgradf$)=0.$

If f has continuous partial derivatives on R$^(3)$ and C is any

circle, then $\backslash$int$\_$C gradf$*$dr$=0.$

If F and G are vector fields, then

curl$($F$*$G$)=$curlF$*$curlG

If S is a sphere and F is a constant vector field, then

$\_($S$)$F$*$dS$=0.$

If F$=\backslash$Pi $+$Qj and P$\_($y$)=$Q$\_($x$)$ in an open region D$,$ then F is

conservative.

$\backslash$int$\_(-$C$)$ f$($x$,$y$)$ds$=-\backslash$int$\_$C f$($x$,$y$)$ds

If F and G are vector fields and divF$=$divG, then F$=$G$.$

The work done by a conservative force field in moving a

particle around a closed path is zero.

If F and G are vector fields, then

curl$($F$+$G$)=$curlF$+$curlG

There is a vector field F such that

curlF$=\backslash$xi $+$yj$+$zk

The area of the region bounded by the positively oriented,

piecewise smooth, simple closed curve C is A$=$o$\backslash$int$\_$C ydx$.$