My working: A basis for the tangent space (at any point) is (1,0), (0,1) so I need only show it for this . df(1,0)=df/dx and df(0,1)=df/dy Is this the right approach? Thanks
yes I was right. A parametrization is conformal if its first fundamental form is of the form $\displaystyle g(du^2+dv^2)$. So just compute its first fundamental form.
Let $\displaystyle A=1+x^2+y^2$. The coordinate vectors are:
$\displaystyle \frac{\partial}{\partial x}=\frac{(2A-4x^2, -4xy, -4x)}{A^2}$
$\displaystyle \frac{\partial}{\partial y}=\frac{(-4xy, 2A-4y^2, -4y)}{A^2}$
It is easy to verify that $\displaystyle \langle \frac{\partial}{\partial x}, \frac{\partial}{\partial x} \rangle=\langle \frac{\partial}{\partial y}, \frac{\partial}{\partial y} \rangle$
And $\displaystyle \langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y} \rangle=0$
I was hoping to solve by using that theorem that a map is conformal if at each point p there is a basis e,e' such that the df(e).df(e')=lambda x e.e' where lamda is the scalar factor. I've just realised I ought to use polar co-ordinates x=rcost, y=rsint. That will simplify things right?
Your theorem is nothing but the statement of "the first fundamental form is of the form $\displaystyle \lambda(dx^2+dy^2)$"
Note that in my post,$\displaystyle \frac{\partial}{\partial x} = df(e_1)$, and $\displaystyle \frac{\partial}{\partial y} = df(e_2)$, given $\displaystyle e_1=(1,0)$ and $\displaystyle e_2=(0,1)$.