[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 2If , 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.