# Velocities - Equation 1.46 of Goldstein 3rd edition

In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein uses the parametrization (equation 1.45')

$$\displaystyle{\vec{r_i}=\vec{r_i}(q_1,q_2, ..., q_n, t)}\tag{1.45'}$$

Where $$\vec{r_i}$$ are the old coordinates and $$\vec{q_k}$$ are the generalized coodinates.

Using the "chain rules" he arrives at the expression (equation 1.46)

$$\displaystyle{\vec{v_i}\equiv \frac{\mathrm{d}\vec{r_i} }{\mathrm{d} t}= \sum_{k}\frac{\partial \vec{r_i}}{\partial q_k} \dot{q_k}+\frac{\partial \vec{r_i}}{\partial t}}\tag{1.46}$$

I didn't understand, what is the difference between $$\displaystyle{\frac{\mathrm{d}\vec{r_i} }{\mathrm{d} t}}$$ and $$\displaystyle{\frac{\partial \vec{r_i}}{\partial t}}$$? I thought they were the same thing, but this way

$$\displaystyle{\sum_{k}\frac{\partial \vec{r_i}}{\partial q_k} \dot{q_k}}$$

vanishes. Is it correct? If it is, why does the sum vanish?

$$\displaystyle{\sum_{k}\frac{\partial \vec{r_i}}{\partial \dot{q_k}}\cdot \vec{\dot{q_k}}}$$
does not vanish. $$\vec{\dot{q_k}}$$ is $${d\vec q_k \over dt}$$.