Free, open-source online mathematics for students, teachers and workers

$$\varphi(u,v)=\left( \begin{array}{c} \cos u\sin v\\ \sin u\sin v\\ \cos v+\log\left(\tan\left(\dfrac{v}{2}\right)\right) + au \end{array} \right)\qquad \begin{array}{l} u\in\mathbb{R}\\ v\in\left(0,\dfrac{\pi}{2}\right) \end{array} $$