energy - Could someone remind me what this means again? $nabla U = pm F$

You know that for a potential function (conservative force/fields) that

$\nabla U = \pm \vec{F}$

In math, we don't have that minus sign, we have only the plus one.

What does it mean if you get rid of the plus sign? I remember the minus signs tells us that the force is always in the opposite direction of the increasing potential function. Does math tell me there exists a potential function such that it's force also increases as potential goes up?

Answer

Mathematically, it just gives a duality with vector fields and scalar fields in multivariable calculus, associated with conservative vector fields and line integrals. As such, the $\pm$ is irrelevant, because it can be absorbed into the force vector. For physics, we take the sign convention to be negative, so that it agrees with the fact that the force is restoring the object it acts on to a lower energy configuration. Note that we could alternatively absorb the negative sign into the potential! It is all a matter of sign convention, and when you define potential and force in physics (as stated above), the negative sign appears in your equation.