Question

Q. The unit sphere S^2 defined by
σ(θ,φ) =(cosθcosφ,cosθsinφ,sinθ)
σ͂(θ,φ) =(-cosθcosφ,-sinθ,-cosθsinφ)

Answer 1

Answer on Question #76062 – Differential Geometry | Topology

Question

The unit sphere $S^2$ defined by:

1) $\sigma(\theta, \varphi) = (\cos\theta\cos\varphi, \cos\theta\sin\varphi, \sin\theta)$

2) $\tilde{\sigma}(\theta, \varphi) = (-\cos\theta\cos\varphi, -\sin\theta, -\cos\theta\sin\varphi)$

Solution

Both functions define a set on the unit sphere, since $x^2 + y^2 + z^2 \equiv 1$ and $\tilde{x}^2 + \tilde{y}^2 + \tilde{z}^2 \equiv 1$. If the domain of the first function is $R^2$, then its image is the whole unit sphere: variable $\frac{\pi}{2} - \varphi$ determines the angle between the $Y$-axis and the radius vector, and variable $\theta$ determines the angle between the projection of the radius vector onto the $XZ$-plane and the $X$-axis. If the domain of the second function is $R^2$, then it also defines the sphere: $\tilde{x} = -x$, $\tilde{y} = -z$, $\tilde{z} = -y$.

Answer

Both these functions define the unit sphere $S^2$, if the domain of them is $R^2$.

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