Let $E\subset\mathbb{R}^n$ be measurable, $\lambda>0,\: x\in\mathbb{R}^n$ and $T\in C^1_c(\mathbb{R}^n;\mathbb{R}^n)$.
Is it correct to write $$ \int_{\lambda E+x}\text{div}_y\:T\:dy=\lambda^{n-1}\int_{E}\text{div}_z\:T(\lambda z+x)\:dz\quad ? $$
The change of variables formula says that for $G$ a diffeomorphism and $f$ (Lebesgue) intergrable $$ \int_{G(\Omega)}f(y)\:dy=\int_{\Omega}(f\circ G)(z)|\text{det}D_zG|\:dz\quad $$ where $D_zG$ is the Jacobian matrix of $G$. I'm thinking of $y=G(z)=\lambda z+x$ and $$\frac{\partial T_i}{\partial y_i}=\frac{\partial T_i}{\partial z_i}\frac{\partial z_i}{\partial y_i}=\lambda^{-1}\frac{\partial T_i}{\partial z_i}$$ so I'm tempted to say yes, but a second opinion would be appreciated.