a) If Y[r]=X[\[Tau][r]] for a smooth parameterization X[t] and t=\[Tau][r] is a smooth scalar function, then the vector (dY/dr)[r] is the scalar multiple of ((d X)/(d t))[\[Tau][r]] given by (dY/dr)[r]=(d t)/dr((d X)/(d t))[\[Tau][r]]=(\[Tau]^\[Prime])[r](X^\[Prime])[\[Tau][r]] For example, write this in coordinates as Y[r]=(Subscript[y,1][r] Subscript[y,2][r] Subscript[y,3][r]) and X[t]=(Subscript[x,1][t] Subscript[x,2][t] Subscript[x,3][t]) where Subscript[y, i][r]=Subscript[x, i][\[Tau][r]].