Define $M:= \{(x,y,z) \in \mathbb R^2: x^2 +y^2 = 1, x+y+z=0 \}$ and define $\Phi_1, \Phi_2:(0,1)\to M$ by
$\Phi_1(t):= (\cos(2\pi t),\sin(2\pi t),-\cos(2\pi t)-\sin(2\pi t)),$
$\Phi_2(t):=(\sqrt{1-t^2},t,-\sqrt{1-t^2}-t)$.
$M$ is a smooth manifold using these charts (proof not needed). Is $M$ an oriented smooth manifold with these charts?
What would be the easiest/most efficient way to find this?
I'm quite stuck with this, here's my idea:
Am I correct in thinking I would find $(\Phi_2^{-1}\circ\Phi_1)(x)$, then $(\Phi_2^{-1}\circ\Phi_1)'(x)$ and if the second value is positive they have the same orientation and it is therefore an oriented manifold?
If so, I'm not entirely sure how to work out those two values, I realize $(\Phi_2^{-1}\circ\Phi_1)(x)$ must be in the $(0,1)$ plane, so thought perhaps $\sqrt{1-t^2}$?
Apologies if my idea doesn't make sense. Any help would be great!