I just spend a while trying to figure out what div(`f`^{i}∂_{i}) is using the definition of div that says d(i_{X}dV)=div(`X`)dV. In the end I just got div(`f`^{i}∂_{i})=∂_{i}`f`^{i}. I was also reminded today that the definition of d`f` is that d`f`(`X`)=`X``f`. I swear all differential geometry amounts to is swapping characters around.