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I am reading Goldstein and I am at a point where I can't follow along. He has started with D'Alembert's Principle and he is showing that Lagrange's equation can be derived from it. He states the chain rule for partial differentiation:

[tex]\frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial \mathbf{r}_i}{\partial q_k}\dot{q}_k+\frac{\partial \mathbf{r}_i}{\partial t}[/tex]

Then he states, by the equation above, that:

[tex]\frac{d}{dt}\frac{d\mathbf{r}_i}{dq_j}=\sum_k \frac{\partial^2 \textbf{r}_i}{\partial q_j \partial q_k}\dot{q}_k+\frac{\partial^2 \mathbf{r}_i}{\partial q_j\partial t}[/tex]

He further states from the first equation that:

[tex]\frac{\partial \mathbf{v}_i}{\partial \dot{q}_j}=\frac{\partial \mathbf{r}_i}{\partial q_j}[/tex]

I have tried to connect the dots but I cannot succeed. Any insight is greatly appreciated. Thanks!

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# Confused About the Chain Rule for Partial Differentiation

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