$\displaystyle \sum_{i,j=1}^{n}\int_{U}a^{ij}u_{x_{i}}\zeta^{2}u_{x_{j}}dx$
$\displaystyle =\sum_{i,j=1}^{n}\int_{U}a^{ij}D_{i}u\zeta^{2}D_{j}u dx$
Can someone please explain to me how we use the chain rule to arrive at
$\displaystyle =\sum_{i,j=1}^{n}\int_{U}\zeta^{2}a^{ij}D_{i}uD_{j}udx+\sum_{i,j=1}^{n}\int_{U}2\zeta a^{ij}uD_{i}u D_{j}\zeta dx$
This just uses the fact that $D_i(u\zeta^2)=\zeta^2D_iu+2u\zeta D_i\zeta.$ Technically this is the product rule, since $D_i(uv)=(D_iu)v+(D_iv)u$, and we have that $v=\zeta^2$, so $D_iv=D_i\zeta^2=2\zeta D_i\zeta$. The latter comes from the composite function derivative rule $(f(g(x)))'=g'(x)f'(g(x))$, we are applying in the case that $g=\zeta$, and $f=x^2$.