I have the following question.
I need to compute the following integral $$\int_A xy+yz+zx \,\,\,dxdydz$$ for $A=\{(x,y,z)\in \mathbb{R}^3: x,y,z\geq 0, x^2+y^2+z^2\leq 1\}$ with the gauss formula.
My Idea was the following:
We have the following situation:
I remark that $\partial A=B_{xy}\cup B_{yz}\cup B_{xz} \cup S$. Now I see that if I take $$F(x,y,z)=(xyz,xyz,xyz)$$ then $div(F)=xy+yz+zx$. In addition I remark that $F$ is zero on $B_{xy}, B_{yz}, B_{xz}$ so it is enough to compute $$\int_S \langle F(x,y,z),\nu(x,y,z)\rangle \,\,\,dS(x,y,z)$$. Since this is a sphere it is clear that the outer normal vector $\nu(x,y,z)=(x,y,z)$ thus we have $$\langle F(x,y,z),\nu(x,y,z)\rangle=x^2yz+xy^2z+xyz^2$$ so I have $$\int_S x^2yz+xy^2z+xyz^2\,\,\,dS(x,y,z)$$Since this situation is symmetric it is enough to compute only one integral. Therefore I wanted to compute $$\int_S x^2yz\,\,\, dS(x,y,z)$$.
But here I have some problems, I don't see how to procede. I thought that maybe we could use the transformation theorem but I'm not sure. Could someone maybe help me how to proceed from here?
So I mean with the transformation theorem I would get $$\int_S x^2yz \,\,\,dS(x,y,z)=\int_0^{\frac{pi}{2}} \int_0^{\frac{pi}{2}} \sin^2(\theta)\cos^2(\rho) \sin(\theta)\sin(\rho)\cos(\theta) \,|sin(\theta)| \,\,\,\,dS(\theta,\rho)$$
Thanks a lot.
Based on further clarifications from you, you are supposed to use gram matrix.
The surface is, $r(\theta, \phi) = (\cos\theta \sin\phi, \sin\theta \sin\phi, \cos\phi), 0 \leq \theta, \phi \leq \pi/2$
Unit normal vector, $\hat n = (\cos\theta \sin\phi, \sin\theta \sin\phi, \cos\phi)$
$dS = \sqrt{\text {det} ~G} ~ d\phi ~d\theta$
To write gram matrix,
$v = (\cos\theta \cos\phi, \sin\theta \cos\phi, - \sin\phi)$ $w = (- \sin\theta \sin\phi, \cos\theta \sin\phi, 0)$
$\displaystyle G=\left(\begin{array}{rrr} \langle v,v \rangle & \langle v,w \rangle \\ \langle w,v \rangle & \langle w,w \rangle \end{array}\right) =\left(\begin{array}{rrr} 1 & 0 \\ 0 & \sin^2\phi \end{array}\right)$
$\text {det } G = \sin^2\phi$
So the integral becomes,
$ \displaystyle \int_0^{\pi/2} \int_0^{\pi/2} (\vec F \cdot \hat n) ~ \sin\phi ~ d\phi ~ d\theta$