Answer
(a) $0$
(b) $curlF$ points in the negative $z-direction$
Work Step by Step
(a) The vector field shown is a 2D vector field of the form $F=Pi+Qj$
We know that
$divF=\frac{∂P}{∂x}+\frac{∂Q}{∂y}$
Here,
$\frac{∂P}{∂x}$ is $0$, since the $x$ components of the vectors are having same length, as we move along the positive $x-direction$.
$\frac{∂Q}{∂y}$, is $0$, since the $y$ components of the vectors are $0$, thus there is change is also $0$.
Which implies that divergence is $0$.
That is, $divF=\frac{∂P}{∂x}+\frac{∂Q}{∂y}=0+0=0$
Hence, $divF$ is $0$.
(b) The vector field shown is a 2D vector field of the form $F=Pi+Qj$
We know that
$curlF=(\frac{∂Q}{∂x}-\frac{∂P}{∂y})k$
Here $\frac{∂Q}{∂x}=0$, since the $y$ components of the vectors are $0$.
$\frac{∂P}{∂y}$ is positive, since the $x$ components of the vectors are increasing as we move upwards.
That is,$curlF=(\frac{∂Q}{∂x}-\frac{∂P}{∂y})k=(0-(+ve))k=(-ve)k$
Hence, $curlF$ points in the negative $z-direction$.
