I know the standard way to parametrize a torus given by Wikipedia. However, I'm trying to parametrize a specific torus, where $R = 2$ and $r = 1$ in Wikipedia's notation by doing this:

(Note that my $r$ is not the same as the notation on Wikipedia. It's simply the radius of a circle).

$\phi(r,\theta) = (r\cos(\theta),r\sin(\theta),\sqrt{1-(2-r)^2}$), where $r \in [1,3]$ and $\theta \in [0,2\pi]$.

This will parametrize the half of the torus above the plane where $z = 0$.

However, when I tried to compute the surface intergral $\int ||D_1\phi \times D_1\phi||$, the integral evaluates to $\frac{1}{2}\bigg[\log(1-r)- 3\log(3-r) \big]$ from $r = 1$ to $r = 3$. This indefinite integral doesn't converge.

What's wrong with my parametrization?

  • $\begingroup$ Is your integral correct? $\endgroup$ – Dylan May 21 '18 at 9:08
  • $\begingroup$ @Dylan Yes, computed it by WolframAlpha. $\endgroup$ – user1691278 May 23 '18 at 2:51

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