This is not a homework problem. I was trying to use the Gaussian linking integral to compute the linking number of the Hopf link and I get stuck at:

$$\int_{0}^{2\pi}\int_{0}^{2\pi}\frac{\cos t-\sin s - \cos t \sin s}{\sqrt{3+2(\cos t - \sin s - \cos t \sin s)}^3}dtds$$

Can anyone suggest some help?

  • 1
    $\begingroup$ Calculating it numerically with Maple, I obtain -9.192212570. If the linking number should be integer, then something to adjust. $\endgroup$
    – user64494
    Jun 9, 2013 at 9:51
  • $\begingroup$ ah yes, I found that I missed a 3 in the exponent of the denominator $\endgroup$
    – wilsonw
    Jun 9, 2013 at 9:56
  • 1
    $\begingroup$ In this case (i.e. with the exponent 3) the integral under consideration equals -12.56637061 which is identified as $ -4 \pi.$ $\endgroup$
    – user64494
    Jun 9, 2013 at 10:31
  • $\begingroup$ Thank you; that's my expected answer, as there should be a factor $\dfrac{1}{4\pi}$ in front of the integral. $\endgroup$
    – wilsonw
    Jun 9, 2013 at 11:01
  • 1
    $\begingroup$ This integral has been exactly calculated with Maple in MaplePrimes. $\endgroup$
    – user64494
    Jul 1, 2013 at 18:38


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