I'm in a differential geometry class and I just attended a review session where the TA gave an example problem about conformal maps on the board:
Find a constant $k$ such that $x(u,v) = (e^{(kv)}cos(v), e^{(kv)}sin(v), e^{(kv)}$ is a conformal parameterization of the cone $z = \sqrt{(x^2+y^2)}$.
So he computed the $E$, $F$ (which is $0$ for any choice of $k$) and $G-$ metric for $x(u,v)$, and set them equal to some scaling factor $λ^2$ times what I assume is the respective $E$ and $G$ of the space being mapped to? $E(x) = λ^2(<1,0><0,1>)$ and $G(x)=λ^2(<1,0><0,1>)$ is what he put.
The $<1,0>$ and $<0,1>$ look like the partial derivatives in respect to $u$ and $v$ of $(u,v)$, but I'm not sure where he got them from. Any ideas?
Also sorry about the lack of typesetting, I don't have latex or anything and I've never posted here before.
$$ x(u,v) = e^{kv}(\cos\ u,\sin\ u,1)$$ Then $$ dx e_1=x_u = e^{kv} (-\sin\ u,\cos\ u,0),\ dx e_2=x_v=k x(u,v) $$ so that $$ E=e^{2kv},\ F=0,\ G=2k^2e^{2kv} $$
That is, if $2k^2=1$ then $$ (dx\ v,dx\ w)=\lambda (v,w)$$ where $$\lambda =E=G=e^{\sqrt{2}v}$$