Why can a force field only be conservative if it is spherically symmetric? I saw in my textbook that a field can only be conservative if it happens to be spherically symmetric. Why is this so? Is there a good proof for this?
 A: No. While a spherically symmetric force field has zero curl and is thus conservative, the converse is not true. As @gandalf61 points out, there can be many, many force-fields that are conservative but not spherically symmetric, I'd urge you to try and construct a couple yourself!
To show why a spherically symmetric field is conervative, here is a very simple one-way proof: a conservative force field $\mathbf{F}$ is one that has zero curl, i.e $$\nabla \times \mathbf{F} = 0.$$
If a force field is spherically symmetric, then it must be written as $\mathbf{F} \equiv F(r)\mathbf{\hat{r}}$. It's now a trivial exercise to show that the curl of any such field is zero, using its definition in spherical coordinates.
A: We know the electromagnetic field is conservative.  A single stationary charge has a spherically symmetric electromagnetic field and is of course conservative; however, the field of two charges (or any number of charges, or moving charges) is also conservative but is not spherically symmetric.
A: A spherically symmetric force field is conservative. But a conservative force field does not have to be spherically symmetric. Example - the force field $\vec F = (x,y,0)$ has cylindrical symmetry but not spherical symmetry.
