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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?

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Actually, after thinking about it more, I now believe we don't need the integral to perform the above steps. All we need to do is use repeated applications of the product rule (but "backwards"), and this doesn't require the use of any integral.

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