TITLE: What is the metric on a cone?
QUESTION [5 upvotes]: I'm trying to learn differential geometry.  I thought a cone would be an easy place to start with calculating a metric, shape operator, what have you.
First of all, by the way, when I say "cone" I mean the map $f: (x,y) \rightarrow (x,y,\sqrt{x^2+y^2})$  Nothing fancy.
So I run through a prescription for getting a metric: $g_{ij} = \partial_i f_k \partial_j f_k$ where repeated indices indicate sum.
I get $G=((2x^2+y^2)\,dx^2+2xy\,dx\,dy+(x^2+2y^2)\,dy^2)/(x^2+y^2)$
Can someone just tell me whether this is right or not, and show me how to correct myself if I have gone wrong?  I can't find this formula or, surprising to me, any other formula anywhere.  I thought this would be a widely discussed example.  Maybe I ought to just stop being cheap and buy a book?

REPLY [2 votes]: This looks pretty much right. You'll know more whether you're doing things right once you compute the second fundamental form as well, because then you can compute the Gaussian curvature, and the Gaussian curvature for a cone is everywhere zero. (Why? Because if you slit it along a line, you can unroll it onto a piece of the flat plane). 
That's a reason that the cone is NOT a great example. A "Monge patch" (the graph of $(x, y) \mapsto (x, y, f(x, y))$ for some $f$) is generally a good deal more informative, because you can get nonzero answers. If you pick $f$ quadratic in $x$ and $y$, everything comes out really nicely.