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Consider the following definition for differentiate ##Y## along ##X##. Write ##Y## in components ##Y = (y^1, \dots, y^n)##, each ##y^i## is a scalar function and differentiate it along ##X## we get ##Xy^i##. Do it for all components we get ##XY##.

I believe there's problem with this definition, which might relate to coordinate dependence. Can somebody explain to me why is this an ill definition? With many different concepts in differential geometry, it gets confusing on what is coordinate dependent and what is not.