contravariant and covariant basis My text book defines covariant (1) and contravariant (2) basis as follows.
$$ \epsilon_i=\frac {\partial x}{\partial q_i} \hat e_x + \frac {\partial y}{\partial q_i} \hat e_y + \frac {\partial z}{\partial q_i} \hat e_z   --(1)$$
$$ \epsilon^i=\frac {\partial q_i}{\partial x} \hat e_x + \frac {\partial q_i}{\partial y} \hat e_y + \frac {\partial q_i}{\partial z} \hat e_z--(2)$$
Then it says, using chain rule
$$ \epsilon^i.\epsilon_j =  \frac {\partial q_i}{\partial x} \frac {\partial x}{\partial q_j} + \frac {\partial q_i}{\partial y} \frac {\partial y}{\partial q_j} + \frac {\partial q_i}{\partial z} \frac {\partial z}{\partial q_j} = \delta^i_j$$
When I see the last equation I think there should be a $3$ factor coming making it $3\delta^i_j$. Please help me to see what's wrong with my understanding. Thanks

By chain rule (the second equality), 
$$\delta _{ij} = \frac{\partial q_i}{\partial q_j} =  \frac {\partial q_i}{\partial x} \frac {\partial x}{\partial q_j} + \frac {\partial q_i}{\partial y} \frac {\partial y}{\partial q_j} + \frac {\partial q_i}{\partial z} \frac {\partial z}{\partial q_j} = \epsilon^i \cdot \epsilon_j $$
