= Boundary integral of position and normal components
{title2=$\int_{\partial\Omega}x_i n_j\,dS=|\Omega|\delta_{ij}$}
For a bounded sufficiently regular region, the <divergence theorem> applied to the vector field $x_i e_j$ gives the displayed identity. Contracting with constant vectors gives $\int_{\partial\Omega}(b\cdot x)(c\cdot n)\,dS=|\Omega|b\cdot c$. Contracting with the <Kronecker delta> or the <Levi-Civita symbol> respectively gives $\int x\cdot n\,dS=3|\Omega|$ and $\int x\times n\,dS=0$ in three dimensions.
Back to article page
