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I've seen in this article that potential energy is defined like this: $U=-\int _{\text{ref}}^r\overset{\rightharpoonup }{F}\cdot d\overset{\rightharpoonup }{r}$.

However I've seen in other text books that potential energy is: $\text{$\Delta $U}$.

What is the difference?

Why is "Delta" used sometimes and sometimes not?

How should I interpret the "Delta" form?

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    $\begingroup$ Fundamentaly, they are the same. $\endgroup$
    – jinawee
    Feb 20, 2014 at 16:38
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    $\begingroup$ Potential energy has a global gauge symmetry in that you can arbitrarily define the point at which is is zero. So $U$ is actually always $\Delta U$ because it's the difference from the point where you've set the zero. Given that $U$ is always $\Delta U$ there's no real confusion in dropping the $\Delta$. Just make sure you're using the same zero when you're comparing different potential energies. $\endgroup$ Feb 20, 2014 at 18:03

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The Delta symbol is often used to describe a "difference". Typically $$\Delta x=x_f-x_i$$ for some parameter $x$. That is, it is the final minus the initial.

In this case, for potential energy, the final and initial states are explicitly mentioned as the limits of the integral. The way it is written here allows you to write the following: $$\Delta U=U(r_f)-U(r_i)=-\int_{\rm{ref}}^{r_f}\vec F\cdot d\vec r+\int_{\rm{ref}}^{r_i}\vec F\cdot d\vec r=-\int_{r_i}^{r_f}\vec F\cdot d\vec r$$ which might be somewhat more clear.

Basically, as you have it written, the integral is taken from an arbitrary reference point which allows you to compare values, but potential energies must always be considered in the context of two separate points.

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