Double integral over the unit sphere.

Question

Let $S$ the unit sphere, and $ \vec a, \vec b $ constant vectors. Prove that : $$ \iint_S \langle \vec x, \vec a \rangle \langle \vec x, \vec b \rangle dO= \frac43π\langle \vec α, \vec b \rangle $$

I have see again this question ,its an exercise of "A First Course in Differential Geometry" of Hsiung. Can anyone give a method of global differential geometry about the solution...

Answer 1

The vector $\vec{x}$ is the outward unit normal to $S$, so since

$$\langle \vec{x},\vec{a}\rangle \langle \vec{x},\vec{b}\rangle = \langle \vec{x}, \vec{a}\langle \vec{x}, \vec{b}\rangle\rangle$$ the divergence theorem gives $$\iint_S \langle \vec{x},\vec{a}\rangle \langle \vec{x},\vec{b}\rangle\, dO = \iiint_D \operatorname{div}(\vec{a}\langle \vec{x}, \vec{b}\rangle)\, dV.$$

Since $\frac{\partial}{\partial x_i} a_i\langle \vec{x}, \vec{b}\rangle = a_ib_i$ for $i = 1, 2, 3$, $$\operatorname{div}(\vec{a}\langle \vec{x}, \vec{b}\rangle) = a_1b_1 + a_2b_2 + a_3 b_3 = \langle \vec{a}, \vec{b}\rangle.$$

Hence

$$\iiint_D \operatorname{div}(\vec{a}\langle \vec{x}, \vec{b}\rangle)\, dV = \iiint_D \langle \vec{a}, \vec{b}\rangle\, dV = \operatorname{vol}(D)\langle \vec{a}, \vec{b}\rangle = \frac{4}{3}\pi \langle \vec{a}, \vec{b}\rangle,$$ as desired.