In the Cartesian coordinates $(x,y)$, I have a vector function $\bar{f}(x)=\hat{x}A\cos(yk)$, where $A$ and $k$ are constants. I make now a 45 degrees rotation (in the same plane) to the new set of Cartesian coordinates $(u,v)$. Since $\sin 45=\cos 45=1/\sqrt{2}$, I know that we will have:
$x=(u-v)/\sqrt{2}$
$y=(u+v)/\sqrt{2}$
Which will give us the function in the new coordinates systems as: $\bar{f}(u,v)=\frac{\hat{u}-\hat{v}}{\sqrt{2}}A\cos[(u+v)k/\sqrt{2}]$.
Is this the correct way to express the function in the new coordinates? Is there any missing scaling factor of any kind?
Thanks.
That looks exactly correct to me. You can check by plugging in $(x, y) = (1, 0)$ or $(x, y = (1,1)$ and comparing to what you get when you plug in $(u, v) = (1, -1)/\sqrt{2}$ (for the first) or $(u, v) = (1, 0)$ (for the second). Since the results are the same, your expression in $uv$-coords is fine.