Question about the del operator under a translation

[DRose87]

Homework Statement

This isn't really a problem. I am just re-reading some section "Classical Mechanics" by John Taylor. I think this belongs in the math section, since my question is mainly about the del operator.

There is just one fragment of one sentence that I want to make sure I am interpreting correctly. For a little bit of background context...we are dealing with an isolated system of two particles. The forces are translationally invariant, depending only on the relative position of the two particles, and conservative.

Next the author says that if we translate the system so that particle 2 is no longer located at the origin,

The next sentence, lifted directly from the book, is what I want to make sure that I understand

2) Attempt at a solution

Is this what the author means when he says that the del operator does not need to be changed when we translate the system.

 

[Charles Link]

A similar thing appears quite often in E&M (electricity and magnetism) where a charge $Q$ located at $x'$ has a potential function given by $V(x)=Q/|x-x'|$. The electric field is given by $E(x)=- \nabla V(x)$ and the gradient only operates on the unprimed coordinate and treats the primed coordinate as a constant. In your mechanics problem, it can be looked at as a translation, but it really is that the location of the center of the potential is a constant and you are just shifting the constant... editing... And yes, I think your calculations are correct, but the simpler way is to just look at it as I just mentioned.