Maxwell's equations for electrostatics

Problem:

Consider the vector field

E(x,y,z) = a(x2 - y2 + z2, z2 - 2xy, 2zy + 2zx),

where a is a constant expressed in the appropriate units.
(a)  Is this field irrotational?
(b)  What is the corresponding charge density?

Solution:

Problem:

In a volume of space V the electric field is

E = c(2x2 -2xy - 2y2)i + c(y2 - 4xy -x2)j,

where c is a constant.
(a)  Verify that this field represents an electrostatic field.
(b)  Determine the charge density ρ in the volume V consistent with this field.

Solution:

Problem:

Can the following vector functions represent static electric fields?   If yes, determine the charge density.  This is not a yes/no question.  You must justify your answer mathematically.
(a)  E = r×(c×r)   (Here c is a constant vector.)
(b)  E = c r r   (Here c is a constant and r = |r|.)

Solution:

Problem:

The electric field in some volume V of space is  E = [2A/(x3y2) – By]i + [2A/(x2y3) – Bx]j.
(a)  Show that the line integral of this field over a close path is zero.
(b)  Which charge density ρ in the volume V is consistent with this field?