I've got one thing left proofing for that one task:
The orientability of a hemisphere with the parameterization $$\psi(\varphi, \theta) = \left(\begin{array}{cc}\cos(\varphi)\,\sin(\theta) \\ \sin(\varphi)\,\sin(\theta)\\ \cos(\theta)\end{array}\right), \quad \varphi \in[0,2\,\pi] \quad \theta\in[0,\dfrac{\pi}{2}]$$
Now, as usual with the unit Normal Vector given by: $$N = \dfrac{\partial_\varphi\psi \times\partial_\theta\psi}{\vert \partial_\varphi\psi \times\partial_\theta\psi \vert} = N = \left(\begin{array}{c}-\cos(\varphi)\,\sin(\theta) \\ -\sin(\varphi)\,\sin(\theta) \\ -\cos(\theta)\end{array}\right) $$ However I don't really can incorporate this in the definition of orientated surfaces one gave us:
A surface is oriented if for each parametrisation $\psi(u,v)$ $$ \pm N\circ\psi = \dfrac{\partial_u\psi \times\partial_v\psi}{\vert \partial_v\psi \times\partial_v\psi \vert}$$
Does this imply I just have to show $$N = \dfrac{\partial_\theta\psi \times\partial_\varphi\psi}{\vert \partial_\theta\psi \times\partial_\varphi\psi \vert} = N = \left(\begin{array}{c}\cos(\varphi)\,\sin(\theta) \\ \sin(\varphi)\,\sin(\theta) \\ \cos(\theta)\end{array}\right) \quad ?$$
I just can't imagine how to cover all parameterizations