- #1

- 98

- 0

## Main Question or Discussion Point

I was reading over a text book that stated the following, where [itex]y(s,t) = v(x(s,t),t)[/itex]

[tex]\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t}[/tex]

and

[tex]\frac{\partial^2y}{\partial t^2} = \frac{\partial^2 v}{\partial x^2}\left ( \frac{\partial x}{\partial t} \right )^2 + 2\frac{\partial^2 v}{\partial x \partial t}\frac{\partial x}{\partial t} + \frac{\partial^2 v}{\partial t^2}+ \frac{\partial v}{\partial x}\frac{\partial^2 x}{\partial t^2}[/tex]

I have been having trouble figuring out how they came to the second statement. My logic proceeds as such, using the product rule:

[tex]\frac{\partial^2y}{\partial t^2} = \frac{\partial}{\partial t}\left [ \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t} \right] = \frac{\partial^2 v}{\partial t \partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial x}\frac{\partial^2 x}{\partial t^2} + \frac{\partial^2 v}{\partial t^2}[/tex]

But certainly my result is not equivalent to the correct one. Where did I go astray? Thanks all!

[tex]\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t}[/tex]

and

[tex]\frac{\partial^2y}{\partial t^2} = \frac{\partial^2 v}{\partial x^2}\left ( \frac{\partial x}{\partial t} \right )^2 + 2\frac{\partial^2 v}{\partial x \partial t}\frac{\partial x}{\partial t} + \frac{\partial^2 v}{\partial t^2}+ \frac{\partial v}{\partial x}\frac{\partial^2 x}{\partial t^2}[/tex]

I have been having trouble figuring out how they came to the second statement. My logic proceeds as such, using the product rule:

[tex]\frac{\partial^2y}{\partial t^2} = \frac{\partial}{\partial t}\left [ \frac{\partial v}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial t} \right] = \frac{\partial^2 v}{\partial t \partial x}\frac{\partial x}{\partial t} + \frac{\partial v}{\partial x}\frac{\partial^2 x}{\partial t^2} + \frac{\partial^2 v}{\partial t^2}[/tex]

But certainly my result is not equivalent to the correct one. Where did I go astray? Thanks all!