This short notation for differential operators is very common in physics since decades. Even better, for instance, while some folks in differential geometry (ie. General relativity, especially the astronomy community) dont write out the operator "any more" but indicates it with a notation like X_{,i} which means \partial_i X.
I don't like that notation, but I always would abbreviate a differential operator :)
How standard is the f \circ (g, h) notation for composition of n-ary functions? I do see it in Wikipedia, but I hadn't come across it myself until looking at Sussman et al. Structure and Interpretation of Classical Mechanics.