Consider the two vector fields $1$ F$$

and $2$ F$$

given below,

$$

$$

$1=2+2$

; $$

$$

$2=2+2($a$)$ Calculate the curl of each of these fields. $(4$ pts$)$

$($b$)$ For each of the fields $1$ F$$

and $2$ F$$

$,$ state if it can be represented as the gradient of a scalar field $,$ and if so$,$

find the field $.(3$ pts$)$

$($c$)$ Calculate the divergence of each of these fields. $(4$ pts$)$

$($d$)$ For each of the fields $1$ F$$

and $2$ F$$

$,$ State whether or not it can be represented as the curl of a vector field H$$

$,$

and if the answer is yes, find H$$

$.(3$ pts$)$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$2$

$1$

$=$

$$

$$

$=$

$$

$$

F

F

$$

$$

$1$ F$$ : $($yes or no$)$

If Yes, $=$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$2$ F$$ : $($yes or no$)$

If Yes, $=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$1$ F$$ : $($yes or no$)$

If Yes, H$$

$=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$2$ F$$ : $($yes or no$)$

If Yes, H$$

$=\_\_\_$