Consider the unit sphere $S^2 = \left\{x\in\mathbb{R}^3 ~ {\large|} ~ |x|=1 \right\}$ and denote the uniform (Lebesgue) measures on the $S^2$ and $[0,1]^2$ by $m_S$ and $m_2$, respectively.

**Question 1:** Is there a map $\phi : [0,1]^2 \to S^2$ such that preserves the uniform measure, i.e., (with a slight abuse of notation) $\phi \circ m_2 = m_S \circ \phi$?

**Question 2:** Is there a way to construct *coordinates* on $(\varphi_1(w),\varphi_2(z)):[0,1]^2 \to S^2$ such that $dm_S(\varphi_1(w),\varphi_2(z)) = dw\,dz$?

**Question 3:** Does such coordinates exist for every higher dimensions?

*Remark:* On $S^1$, this is trivial by $\varphi (t) = e^{2\pi i t}$, up to a constant.