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Problem 39

Source:

July 1, 2010
Gausscalculusintegrationtrigonometry

Problem Statement

Calculate the Gauss integral
∮(dA→,dB→,A→−B→)∣A→−B→∣3\oint\frac{(d\overrightarrow{A},d\overrightarrow{B},\overrightarrow{A}-\overrightarrow{B})}{|\overrightarrow{A}-\overrightarrow{B}|^3}
where A→\overrightarrow{A} runs along the curve x=cos⁡αx=\cos\alpha, y=sin⁡αy=\sin\alpha, z=0z=0, and B→\overrightarrow{B} along the curve x=2cos⁡2βx=2\cos^2\beta, y=12sin⁡βy=\frac12\sin\beta, z=sin⁡2βz=\sin2\beta.
Note: that ∮\oint was supposed to be oiint (i.e. ∬\iint with a circle) but the command does not work on AoPS.
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