The field-theoretic version of (closed & exact) differential forms uses the variational bicomplex and jet bundles, see e.g. Refs. 1-3.

References:

1. P.J. Olver, Applications of Lie Groups to Differential Equations, 1993.

2. I. Anderson, Introduction to variational bicomplex, Contemp. Math. 132 (1992) 51.

3. G. Barnich, F. Brandt & M. Henneaux, Local BRST cohomology, Phys. Rep. 338 (2000) 439.

The field-theoretic version of (closed & exact) differential forms uses the variational bicomplex and jet bundles, see e.g. Refs. 1-3.

References:

1. P.J. Olver, Applications of Lie Groups to Differential Equations, 1993.

2. I. Anderson, Introduction to variational bicomplex, Contemp. Math. 132 (1992) 51.

3. G. Barnich, F. Brandt & M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rep. 338 (2000) 439, arXiv:hep-th/0002245.