[Updated on 20-6-2011 by adding a lemma, in the hope that this will make things more transparent. ]
Inspired by Wonging’s result on integrating products of dot products, we have another Wongting-type theorem:
Theorem 1 (Wongting’s theorem, 2011) For ,
where is the volume of the unit ball in .
I intend to generalize it further, but since is not defined in , , it is inappropriate to integrate on , so we have to find other space to integrate the corresponding cross product of cross product. I plan to do it later.
To prove the theorem, we use the following lemma.
Lemma 2 If , then
Proof: We can w.l.o.g. assume . Let and let forms a positive orthnormal basis. Note that any component of which is parallel to will not contribute to both sides of the equation. So we can assume , thus . Similarly , so as .
Note that this lemma can actually be seen geometrically.
Proof of Wongting’s theorem:
By the linearity of both sides of the expression of Wongting’s theorem and the anti-symmetric property of , we can assume that
By the lemma, the integrand on LHS of the statement in the theorem is .
By the antisymmetry of the funcions and upon reflection of the -plane, the first two components integrate to be zero over the sphere. For the third component,
Thus . Clearly , so we get the result.