Find the curl of the fields shown on the image below.

a) A curl is the vector derivative of a vector field. It can be denoted as 
vec grad xx vec F , where vec F is the vector field.
The curl is calculated as three-dimensional determinant:
i           j            k
d/dx     d/dy      d/dz
F_x      F_y      F_z
This determinant equals the vector quantity
((dF_z)/dy - (dF_y)/(dz)) veci - ((dF_z)/(dx) - (dF_x)/(dz))vecj + ((dF_y)/(dx) - (dF_x)/(dy))veck .
To find curl of the given field, let's first find all the required partial derivatives:
(dF_y)/(dx) = y^2z
(dF_x)/(dy) = x^2z
(dF_z)/(dy) = xz^2
(dF_y)/(dz) = xy^2
 
(dF_z)/(dx) = yz^2
(dF_x)/(dz) = x^2y
Substituting these into the expression above, we get
vec grad xx vecF = x(z^2 - y^2) veci + y(x^2 - z^2) vecj + z(y^2 - x^2) veck .
This is the curl of the given vector field.
b) Follow the same procedure to find the curl of this given vector field, as well.
These are the partial derivatives:
(dF_y)/(dx) = y^2
(dF_x)/(dy) = -xsiny
(dF_z)/(dy) = 0
(dF_y)/(dz) = 0
(dF_z)/(dx) = 0
(dF_x)/(dz) = 0
So the only component of the curl of this field that is non-zero is the z-component:
vecgrad xx vecF = (y^2+xsiny) veck .