I am reading a derivation of the Euler Lagrange equations. They defined coordinate charts φ: U -> R^n where x -> φ(x) = (x1, x2, ... xn). then they said for the one form dφ: TU -> TR^n ~= R^n X R^n where (x,y) -> dφ(x,y) = (x1, ..., xn, y1, ..., yn). what im confused about is that when i learned differential forms, i learned that they are always a map from the tangent bundle to R. they return real numbers. i don't understand how they got dφ: TU -> TR^n ~= R^n X R^n. can someone help to clarify this for me? thanks.