Welcome to our community

Be a part of something great, join today!

Chain rule of differentials

dwsmith

Well-known member
Feb 1, 2012
1,673
Take \(U(\eta) = u(x - ct)\) and the wave equation \(u_{tt} - u_{xx} = \sin(u)\). Then making the transformation, we have
\[
(1 - c^2)U_{\eta\eta} = \sin(u).
\]
My question is the chain rule on the differential.
\[
U_{\eta} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial\eta} + \frac{\partial u}{\partial t} \frac{\partial t}{\partial\eta}
\]
but this doesn't seem to work out correctly.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Re: chain rule of differentials

If I take \(\eta = x - ct\), then
\[
\partial_x = \partial_{\eta}\frac{\partial\eta}{\partial x} = \partial_{\eta}
\]
and
\[
\partial_t = \partial_{\eta}\frac{\partial\eta}{\partial t} = -c\partial_{\eta}
\]
Therefore, the transformation yields
\[
u_{\eta\eta}(c^2 - 1) = \sin(u).
\]
How do am I suppose to get \(1 - c^2\)?