How to Understand and Solve the Chain Rule Problem in Calculus?

[Another1]

\[ \frac{\partial \dot{r}}{\partial \dot{q_k}} = \frac{\partial r}{\partial q_k} \]
where
\[ r = r(q_1,...,q_n,t \]

solution

\[ \frac{dr }{dt } = \frac{\partial r}{\partial t} + \sum_{i} \frac{\partial r}{\partial q_i}\frac{\partial q_i}{\partial t}\]
\[ \dot{r} = \frac{\partial r}{\partial t} + \sum_{i} \frac{\partial r}{\partial q_i}\dot{q_i}\]

let
\[ \frac{\partial\dot{r}}{\partial \dot{q_k}} = \frac{\partial^2r}{\partial\dot{q_k} \partial t} +\frac{\partial}{\partial \dot{q_k}} ( \frac{\partial r}{\partial q_1} \dot{q_1} + ... + \frac{\partial r}{\partial q_k} \dot{q_k} + ... + \frac{\partial r}{\partial q_n} \dot{q_n} ) \]

I think
\[\frac{\partial^2r}{\partial\dot{q_k} \partial t} = 0 \]
and
\[ \frac{\partial}{\partial \dot{q_k}} ( \frac{\partial r}{\partial q_n} \dot{q_n}) = 0\]for k not equal to n

 

[HOI]

What does $\dot{r}$ mean and how is it different from $r$?

 

[Another1]

What does $\dot{r}$ mean and how is it different from $r$?

$\dot{r} $ mean full derivative of r by dt because \[ r=r(q_1,...,q_n,t) \] and \[ q_n = q_n(t) \]
any $q_n$ as function of time so $\dot{r}$is formed by taking the derivative with respect to dt for $( q_1,...,q_n,t )$