Property of the difference quotient in Evans(Partial Differential Equations)

Question

Why holds the property of the difference quotient in Evans(Partial Differential Equations) \begin{equation} \int_{U}v D_k^{-h}dx = -\int_U w D_k^hv dx \end{equation} for $v,w \in H^{1}_0(U)$ (16) in Evan's book, page 311. Where the difference quotient is given by \begin{equation} D_k^hu(x) = \dfrac{u(x+he_k) -u(x)}{h} \ (h \in \mathbb{R}, h\neq 0). \end{equation}

Answer 1

It must be presumed that in these difference quotients, one considers $u(x)$ to be zero when $x\notin u$. Then all the integrals can be taken over $\mathbb{R}^n$.

Now write $$h\int vD_k^{-h}\,dx=\int v(x)u(x-he_k)\,dx-\int v(x)u(x)\,dx$$ and change variables in the former integral on the right, replacing $x$ by $x+he_k$. Then collect the two integrals into one again, and voilà.