# Understanding the first variational formula

I am trying to understand the formula in section 1 of this nLab page.

It says if $$L$$ is an $$n$$-form on the jet bundle $$J\rightarrow X$$ of a smooth bundle $$E\rightarrow X$$, then we can write $$dL = E-d_H\theta$$ where $$E$$ is a source form, and $$\theta$$ is some form on $$J$$, and $$d_H$$ is the horizontal differential.

However, as I understand it, from the classical standpoint, getting this expression from varying the action integral (i..e the integral of the Lagrangian, not just the Lagrangian itself) involves the following two steps:

1. integrating by parts; and,
2. using Stokes' theorem to write the resulting boundary term as the integral of a total derivative, to get $$\theta$$.

The above steps require us to work with the action integral, not just the Lagrangian. Consequently, isn't it incorrect to say that $$dL = E - d_H\theta$$ holds as an on-the-nose equality of differential forms (on $$J$$), and instead we should say these have equal integral over the base space $$X$$ for each possible field configuration, when contracted along any field variation?