Confused About the Chain Rule for Partial Differentiation

[coca-cola]

Hey all,

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:
$$\frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial \mathbf{r}_i}{\partial q_k}\dot{q}_k+\frac{\partial \mathbf{r}_i}{\partial t}$$

Then he states, by the equation above, that:
$$\frac{d}{dt}\frac{d\mathbf{r}_i}{dq_j}=\sum_k \frac{\partial^2 \textbf{r}_i}{\partial q_j \partial q_k}\dot{q}_k+\frac{\partial^2 \mathbf{r}_i}{\partial q_j\partial t}$$

He further states from the first equation that:
$$\frac{\partial \mathbf{v}_i}{\partial \dot{q}_j}=\frac{\partial \mathbf{r}_i}{\partial q_j}$$

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

 

[Mark44]

Try using \mathbf{} instead of textbf. Also, you're missing a \ on the frac command in your second LaTeX line.
Here is a snippet using mathbf: $\frac{d}{dt}\frac{d\mathbf{r_i}}{dq_j}$

Unrendered, this is # #\frac{d}{dt}\frac{d\mathbf{r_i}}{dq_j}# # (spaces added between # chars to prevent rendering). If you fix your LaTeX code, I'll take a look. Otherwise, it's too complicated to try to figure out what you wrote.

 

[Chestermiller]

Also, on your final tex, the slash should be in the opposite direction, i.e., /tex.

It is a good practice to preview your LaTex equations while they are partially under construction so that you can spot errors early on and correct them.

 

[SteamKing]

Hey all,

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:
$$\frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial \textbf{r}_i}{\partial q_k}\dot{q}_k+\frac{\partial \textbf{r}_i}{\partial t}$$

Then he states, by the equation above, that:
$$\frac{d}{dt}\frac{d\textbf{r}_i}{dq_j}=\sum_k \frac{\partial^2 \textbf{r}_i}{\partial q_j \partial q_k}\dot{q}_k+\frac{\partial^2 \textbf{r}_i}{\partial q_j\partial t}$$

He further states from the first equation that:
$$\frac{\partial \textbf{v}_i}{\partial \dot{q}_j}=\frac{\partial \textbf{r}_i}{\partial q_j}$$

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

Equations from OP fixed above. When doing heavy Latex work, always check your post by hitting the Preview button to make sure everything is correct.

 

[coca-cola]

Hey, just checked back in. My apologies guys, I don't post often. I didn't even see the preview option. I'll keep that in mind. Thanks for the formatting help!