This is a question just out of interest to know the power of integration by parts. There are various level of integration by parts. What are some of the most general form of integration by parts? I have encountered it very often in PDE's. I look forward to gaining more insights on it. Thank you for your ideas, help and discussions.
All versions of integration by parts that I have seen boil down to two things.
The only other ingredient that is sometimes needed is some basic housecleaning coming from Riemannian and/or differential geometry: things like how covariant or coordinate partial derivatives relate to the exterior derivative, and how to write the divergence of a vector field as equivalently the exterior derivative of its dual $n-1$ form.