Vanishing Mean of a Derivative and Integration by Parts on the Torus

In the setting of The Flat Torus: Standing Notation. For $u\in C_{\mathrm{per}}$ the restriction $u|_{Q}$ lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i), and we abbreviate

$$\int_{\mathbb{T}^{n}}u\,dx=\int_{\mathbb{T}^{n}}u|_{Q}\,dx .$$

Recall from Elementary Properties of Lattice-Periodic Functions (ii) that $\partial_{i}g\in C_{\mathrm{per}}$ for $g\in C^{1}_{\mathrm{per}}$ and $i\in[n]$, and from Elementary Properties of Lattice-Periodic Functions (i) that $C_{\mathrm{per}}$ is closed under pointwise products, so that all four integrands below lie in $C_{\mathrm{per}}$ and the abbreviation applies to them. Then the following hold.

1. (Vanishing mean of a derivative)(i) Let $g\in C^{1}_{\mathrm{per}}$ and let $i\in[n]$. Then

$$\int_{\mathbb{T}^{n}}\partial_{i}g\,dx=0 .$$

2. (Integration by parts)(ii) Let $g,h\in C^{1}_{\mathrm{per}}$ and let $i\in[n]$. Then

$$\int_{\mathbb{T}^{n}}(\partial_{i}g)\,h\,dx=-\int_{\mathbb{T}^{n}}g\,(\partial_{i}h)\,dx .$$