/sci/ - Science & Math

question about the d'alembert operator, what is the difference between a downstairs differential and an upstairs differential? is it literally just the derivative acting on a contravariant or covariant vector? does it matter that the operator is defined as pic related instead of the other way around?

if it is just the derivative acting on a (co/contra)variant vector, what would happen if i did a covariant derivative on a contravariant vector, would it just be undefined? zero?