Given $\phi: [0, 2\pi ]^2 \to \mathbb{R}^4$, $\phi (\alpha , \beta) = (x(\alpha , \beta), y(\alpha , \beta), z(\alpha , \beta), u(\alpha , \beta))$, where:
- $x(\alpha , \beta) = (4 + \cos \alpha) \cos \beta$
- $y(\alpha , \beta) = (4 + \cos \alpha) \sin \beta$
- $z(\alpha , \beta) = \sin \alpha \cos \frac{\beta}{2}$
- $u(\alpha , \beta) = \sin \alpha \sin \frac{\beta}{2}$
I want to check if the image of $\phi$ is a 2d manifold and find it's atlas.
Tho check whether it is a manifold it enaugh that I calculate partial derivatives of $x$, $y$, $z$, and $u$ with respect to $\alpha$ (in one row) and $\beta$ (in another row) and then having them in matrix I will check that both vectors are independent? How do I find it's atlas?