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He says for two vector fields, ##X## and ##Y##, their commutator can be defined by its action on a scalar function, ##f(x^{\mu})##:$$[X,Y](f) \equiv X(Y(f)) - Y(X(f))$$This is the first time I’m seeing a vector “acting on a function”. What exactly is the nature of this action? I’m very familiar with a vector’s action on a one-form, so my only guess is that this is a shorthand way of saying “a vector acting on the

*gradient*of a function”, so then ##X(Y(f))## would really be ##X ( \text{d} ( Y ( \text{d} f ) ) )##. Is that right? Or am I way off?