Unable to spot the mistake in change of variables in $\mathbb{R}^N$

Question

Let $Q \subset \mathbb{R}^N$ be the cube of side length 2 centred on the origin, $Q_+, Q_0$ and $Q_-$ be the upper half, equatorial and lower halves of the cube. Let $u \in C(Q_+)$ and $\varphi \in C^{1}(Q_-)$. I write any $x \in \mathbb{R}^N = (x_1,x_2, \dots, x_N) = (x',x_N)$. Suppose that the following expression is finite. $$\int_{Q_-}u(y',-y_{N}) \frac{\partial\varphi}{\partial y_i}(y)dy$$ I want to change the domain of the integral to $Q_+$. So I consider the diffeomorphism $H: Q_+ \to Q_-; H(x) = y =(x'-x_N)$. Doing the algebra gives me the following $$-\int_{Q_+}u(x)\frac{\partial \varphi}{\partial x_i}(x',-x_N)dx$$ There does not seem to be a negative sign in the reference. Can anyone find the mistake?