球面の媒介変数表示

$ \forall x,y,z\in\R;x^2+y^2+z^2=1\iff\exist!\theta\in[0,\pi\lbrack\exist!\phi\in[0,2\pi\lbrack;\begin{dcases}x=\sin\theta\cos\phi\\y=\sin\theta\sin\phi\\z=\cos\theta\end{dcases}

証明

$ x^2+y^2+z^2=1

$ \iff \exist! s\in[0,1];

$ \begin{dcases}s^2=x^2+y^2\\s^2+z^2=1\end{dcases}

$ \iff \exist! s\in\R\exist!\theta\in[0,\pi];

$ \begin{dcases}s^2=x^2+y^2\\s=\sin\theta\\z=\cos\theta\end{dcases}

$ \iff \exist! s\in\R\exist!\theta\in[0,\pi]\exist!\phi\in[0,2\pi\lbrack;

$ \begin{dcases}x=s\cos\phi\\y=s\sin\phi\\s=\sin\theta\\z=\cos\theta\end{dcases}

$ \underline{\iff\exist!\theta\in[0,\pi\lbrack\exist!\phi\in[0,2\pi\lbrack;\begin{dcases}x=\sin\theta\cos\phi\\y=\sin\theta\sin\phi\\z=\cos\theta\end{dcases}\quad}_\blacksquare