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You have an attractive potential like the gravitational one.

-∫F.dr = U2 - U1 = ΔU

If you go from U1 to U2, an external agent has to provide positive work against the direction of the Force Field. But your gravitational potential has to deliver negative work in order to raise its potential to U2. The F in the equation is gravitational force. So an attractive potential does negative work to lower its potential.

U2-U1 is positive, in combination with the minus sign, your work by the gravitational force is negative.

How to rewrite your equation if you want F to be the Force exerted by the external agent?

∫Fext.dr = U2 - U1 = ΔU

I would say drop the minus sign, but the general form for a conservative force is F = -∇U

Is my error of thought because of the fact that the F in F = -∇U is your gravitational force due to your gravitational potential and so F = -∇U is not valid for you're external agent? Because the your external agent is in the opposite direction of your force.

Now

If you go back from U2 to U1

-∫F.dr = U1-U2

Right side is negative, so your work done by the gravitational field is positive. So an attractive potential does positive work in order to lower its potential.

But what does your external agent do in this case? Now the external force is aligned with the force field, so I guess that the external agent also delivers positive work?

So equation for the external force:

-∫Fext.dr = U1-U2 = ΔU

So the equations for your field -∫F.dr = ΔU, always stays the same.

But the equations for your external agent differ by a sign.

Is this normal?

But I thought work done by external agent = -1 * work by the field you're acting in?

Am I thinking too much. Semantics, I hate it.

Excuse me for my horrible paint drawing. :)

Also, If you choose youre Potential to be zero at infinity. An attractive potential is always negative and a repulsive potential is always positive right?