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 Sci Advisor HW Helper PF Gold P: 4,771 So F*p takes a vector X at p (derivation on the germs of smooth function at p) and sends it to a vector F*pX at F(p) (derivation on the germs of smooth function at F(p)). So what is this vector F*pX then? How does it act on a germ f at F(p)? This is what the formula is telling you: it says (F*pX)(f) is just X(F*f). And this makes sense, since F*f =f o F is indeed a germ of functions at p.