Why do we need in mathematical physics (as I know in English it is called Partial Differential Equations) orthogonal transformations coordinates?
(for example, the heat equation and the wave equation)
I know that the Laplacian is invariant under such transformations. (In other words it does not change its form).
However, what's the difference?
let's say I have an equation:
$u'_t=\Delta u$, and condition $u\bigr|_{t=0}=e^{-(x+y-z)^2}$, and task is to find $u(t,x,y,z)$
First of all, I should do change of variables: $\xi=\xi(x,y,z) = x + y - z $
(And you can see that such change of variable can't be orthogonal already! Because $\{b_{ij}\}=A^{t}\cdot A \neq E$, since we have $b_{11}=1\cdot1+1\cdot1+(-1)\cdot(-1)=3\neq 1$)
By the way, everything is solving easily. Let's do it.
And after change of variables we should try to find solution as $\hat u(\xi,t)$
then I say $u'_t = \Delta_{x,y,z}\hat u(\xi(x,y,z),t) =3 \hat u_{\xi\xi}$.
And that's why $\hat u (\xi,t)$ is a solution to equation $u'_t = 3 \hat u_{\xi\xi} $ with condition $\hat u\bigr|_{t=0}=e^{-\xi^2}$
That can be easily solved.
But my teacher said that we should use only orthogonal transformations!
For what?
I say nothing about about $\eta, \zeta$.
You can say: "ask this teacher, nobody knows what he meant!". But I will not see him next two weeks.