Find the divergence of the fields shown on the attached image.

a) Divergence of a vector field is a scalar quantity that represents how the field spreads out, or "diverges", in different directions. It is usually denoted as
vecgrad*vecF and is calculated as
vecgrad * vecF = (dF_x)/(dx) + (dF_y)/(dy) + (dF_z)/(dz) .
In the given vector field, the components are
F_x = x , so (dF_x)/(dx) = 1
F_y = y^3z^2 , so (dF_y)/(dy) = 3y^2z^2
F_z = xz^3 , so (dF_z)/(dz) = 3xz^2 .
Thus, the divergence of the given vector field is
vec grad * vecF = 1 + 3y^2z^2 + 3xz^2 .
b) The divergence of this vector field can be calculated the same way. Here,
(dF_x)/(dx) = cosy
(dF_y)/(dy) = 2xy
and (dF_z)/(dz) = 0
So the divergence is
vec grad * vecF = cosy + 2xy .
 
http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx