I am very familiar with the concept of a potential function, and potential energy, from calculus-based physics.

For instance, if we have the familiar force field $\mathbf{F} = -mg \,\mathbf{j}$, then a potential function is given by $U = mgy + C$. (Since potential energy is relative, we have an infinite number of potential functions.)

Notice that the gradient of the potential function is the negative of the force field: $$\nabla U = \nabla(mgy + C) = mg \,\mathbf{j} = -\mathbf{F}.$$

That was perfectly fine with me. But now in vector calculus, I am reading that the potential function $f$ of a vector function $\mathbf{F}$ is such that $\nabla f = \mathbf{F}$. A negative sign appears to have been lost when migrating from physics to calculus.

It seems confusing to call $f$ a "potential function", since it cannot be interpreted as potential energy in the real world. So why is the calculus nomenclature as it is (i.e., why not call this something else and then say the potential function is the negative of it)?


Question: Why does George Mcfly employ Biff to wax his car after he had previously tried to rape his wife?
Answer: 
It's not clear that Biff was trying to rape Loraine.  Although he was clearly taking liberties he should not have.
McFly subdued Biff in the fight at the dance.  After that, Biff learned to take his place behind McFly.  The car-waxing scene was really just to prove this.
30 years had passed.  The new, wiser McFly was undoubtedly able to forgive Biff--especially with Biff in his new subordinate role.