Can anyone give me some help with this problem? I have never worked with such kind of problem and I am a bit confused, specially about what he means with "using symmetry".
If $B_1=B(0,1)$ in $\mathbb{R}^n$, using symmetry, show that:
\begin{align} \int_{\partial B_{1}} x_j \mathrm{d}S = \int_{\partial B_{1}} x_j x_k \mathrm{d}S = 0, \,\,\,\mathrm{for} \,\,\,j\neq k \end{align} and \begin{align} \int_{\partial B_{1}} x_k^{2} \mathrm{d}S = n ^{-1}\int_{\partial B_{1}} \left[ \sum_{j=1}^{n} x_j^{2}\right] \mathrm{d}S = \int_{\partial B_{1}} 1 \mathrm{d}S. \end{align}
Thanks in advance.
For the first part, split $\partial B_1$ into $\partial B_1^+:=\{x\in \partial B_1|x_j\ge0\}$ and $\partial B_1^-:=\{x\in \partial B_1|x_j\lt0\}$, and note that$$\int_{\partial B_1^\color{blue}{-}}x_j\mathrm{d}S=-\int_{\partial B_1^\color{blue}{+}}x_j\mathrm{d}S.$$When we seek to evaluate $\int_{\partial B_1}x_jx_k\mathrm{d}S$ for $j\ne k$, we must have $n\ge2$, so we can perform a similar argument from cancellation, this time splitting based on the sign of $x_j/x_k$. (We don't need to worry too much about the zero-measure locus $x_k=0$, as it doesn't contribute to the integral). The final calculation uses the fact that $\int_{\partial B_1}x_j^2dS$ is $j$-independent, because $\partial B_1$ is invariant under permuting the Cartesian coordinates, so e.g. $\int_{\partial B_1}\left(x_j^2-x_\ell^2\right)dS=0$.