Integration by Parts on Euclidean Space Against a Compactly Supported Function of Class $C^{1}$

Adopt Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation and Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, and fix a dimension $q$. A function $f:\mathbb{R}^{q}\to\mathbb{R}$ is integrable if it is integrable with respect to $\lambda_{q}$, and compactly supported refers to the topology of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (ii); for $f,g:\mathbb{R}^{q}\to\mathbb{R}$ the product $fg$ is the function $x\mapsto f(x)\,g(x)$. Let $i\in[q]$.

1. (A partial derivative integrates to zero)(i) Let $h:\mathbb{R}^{q}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{q}$ and compactly supported. Then $\partial_{i}h$ is continuous on $\mathbb{R}^{q}$, compactly supported and integrable, and

$$\int_{\mathbb{R}^{q}}\partial_{i}h\,d\lambda_{q}=0 .$$

2. (Integration by parts)(ii) Let $f,g:\mathbb{R}^{q}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{q}$, with $g$ compactly supported. Then $f\,\partial_{i}g$ and $(\partial_{i}f)\,g$ are continuous on $\mathbb{R}^{q}$, compactly supported and integrable, and

$$\int_{\mathbb{R}^{q}}f\,\partial_{i}g\,d\lambda_{q}=-\int_{\mathbb{R}^{q}}(\partial_{i}f)\,g\,d\lambda_{q} .$$