Change of basis in tangent space of $C\equiv x^2+y^2=z^2, \, z>0$

Question

Let $C$ be the smooth manifold defined by $x^2+y^2=z^2$ for $z>0$ and consider the following parametrizations: $$\begin{array}{rcll} \Phi:&\mathbb R^+\times (0,2\pi)&\longrightarrow &C\\ &(t,\theta)&\longmapsto &\Phi(t,\theta)=(t\cos\theta,t\sin\theta,t) \end{array} $$ $$\begin{array}{rcll} \Psi:&\mathbb R^2&\longrightarrow &C\\ &(u,v)&\longmapsto &\Psi(u,v)=(u,v,\sqrt{u^2+v^2}) \end{array} $$ It is clear that $\{\Phi_1=\partial \Phi/\partial t, \Phi_2=\partial \Phi/\partial \theta\}$ and $\{\Psi_1=\partial\Psi/\partial u, \Psi_2=\partial \Psi/\partial v\}$ define two basis of the tangent space of $C$ on each point: $$\Phi_1(t,\theta)=(\cos\theta,\sin\theta,1)=\Big(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}},1\Big),\quad\Phi_2(t,\theta)=(-t\sin\theta,t\cos\theta,0)=(-y,x,0)$$ $$\Psi_1(u,v)=\Big(1,0,\frac{u}{\sqrt{u^2+v^2}}\Big),\quad \Psi_2(u,v)=\Big(0,1,\frac{v}{\sqrt{u^2+v^2}}\Big)$$ These basis must be related by a $2\times 2$ matrix but I am not able to find it. Any help?

Answer 1

Between the two coordinated systems it is the polar change $$\left[\begin{array}{cc}u\\\\v\end{array}\right]\longmapsto \left[\begin{array}{cc}t\\\\\theta\end{array}\right]= \left[\begin{array}{cc}\sqrt{u^2+v^2}\\\\\arctan\left(\dfrac{v}{u}\right)\end{array}\right]$$ Call this change $\lambda$.

Then $\Psi=\Phi\circ\lambda$ serves to communicate both systems.

By the chain's rule $J\Psi|_p=J\Psi|_{\lambda(p)}\cdot J\lambda|_p$, which explicitly is $$ \left[\begin{array}{cc}1&0\\\\0&1\\\\\dfrac{u}{\sqrt{u^2+v^2}}&\dfrac{v}{\sqrt{u^2+v^2}}\end{array}\right] = \left[\begin{array}{cc}\cos\theta&-t\sin\theta\\\\\sin\theta&t\cos\theta\\\\1&0\end{array}\right]\cdot \left[\begin{array}{cc}\dfrac{u}{\sqrt{u^2+v^2}}&\dfrac{v}{\sqrt{u^2+v^2}}\\\\ \dfrac{-v}{u^2+v^2}&\dfrac{u}{u^2+v^2}\end{array}\right]\qquad (*)$$ You should check how up on the subbings $$t=\sqrt{u^2+v^2}\quad\mbox{and}\quad \theta=\arctan\left({\dfrac{v}{u}}\right),$$ or $$u=t\cos\theta\quad\mbox{and}\quad v=t\sin\theta,$$ the relation $(*)$ is preserved.