Example

When $X = \mathbf{R}^n$, $\xi$ is the usual orientation of $\mathbf{R}^n$ and $A$ is a closed ball of radius $> 0$, the orientation of the sphere $\partial A$ defined by $\xi$ is the canonical orientation (10.2.8, b)).

11.2.2. Let $\omega$ be a twisted differential form of degree $p$ on $X$ with values in $E$ (10.4.1); the inverse image $i^*(\omega)$ of $\omega$ by the oriented morphism $i : \partial A \to X$ is denoted by $\omega|_{\partial A}$ and is called the form induced by $\omega$ on $\partial A$ (cf. 10.4.3).

11.2.3. Let us make one of the following two hypotheses:

(i) $\omega$ is a twisted differential form of degree $n-1$ on $X$, with values in $E$; (ii) $X$ is oriented, $\partial A$ is provided with the corresponding orientation (11.2.1) and $\omega$ is a differential form of degree $n-1$ on $X$ with values in $E$.

Suppose in addition that $\omega$ is of class $C^1$ and that the intersection of $A$ and the support of $\omega$ is compact. The exterior differential $d\omega$ of $\omega$ is continuous (8.3.5 and 10.3.4).

The characteristic function of $A$ is essentially integrable for the vector measure defined by $d\omega$ on $X$ and the differential form $\omega|_{\partial A}$ of degree $n - 1$ on $\partial A$ is integrable (10.4.3 and 10.4.4). One has

11.2.4. Let $\alpha$ be a twisted differential form of degree $n$ on $X$ with values in $E$, with compact support, and of class $C^1$. In order that there exist a twisted differential form $\omega$ of degree $n - 1$ on $X$, with values in $E$, with compact support and of class $C^1$, such that $\alpha = d\omega$, it is necessary that $\int_X \alpha = 0$; if $X$ is connected, this condition is sufficient; if in addition $\alpha$ is of class $C^k$ ($k \leq r - 1, k \neq \omega$), one can choose $\omega$ of class $C^k$.

11.2.5. Suppose that $X$ is the real variety underlying $\mathbf{C}$, that $E$ is a complex Banach space and that $A$ is a compact piece of $X$. Equip $X$ with the orientation defined by its complex structure (10.2.7). Let $f$ be a continuous mapping from $A$ into $E$ whose restriction to the interior $\mathring{A}$ of $A$ is holomorphic. Let $dz$ denote the differential of the injection $z : \partial A \to \mathbf{C}$. The form $f.dz$, product of $f$ and $dz$, is a differential form of degree 1 on $\partial A$, with values in $E$, of class $C^0$, and one has:

When $f$ extends to a holomorphic mapping, still denoted $f$, of an open neighbourhood $U$ of $A$, with values in $E$, the differential form $f.dz$ (where $z$ denotes this time the canonical injection of $U$ into $\mathbf{C}$) is of class $C^\infty$ on $U$ and its exterior differential is zero.

11.2.6 (Derivative of an integral). Suppose $X$ is oriented. Let $Y$ be a variety of class $C^r$, and let $\alpha$ be a differential form of degree $n$ on $Y$, with values in $E$, of class $C^1$, and with compact support. Let $I$ be an open subset of $\mathbf{R}$ containing 0 and let $g : I \times X \to Y$ be a morphism of class $C^r$; for $t \in I$, denote by $g_t$ the mapping $x \mapsto g(t, x)$ from $X$ into $Y$. Denote by $\psi$ the morphism from $X$ into $T(Y)$ defined by $\psi(x) = T_{(0, x)}(g)(1, 0)$; it is a lifting of $g_0$ (8.6.5). Suppose that the restriction of $\mathrm{pr}_1 : I \times A \to I$ to the intersection of $I \times A$ and the support of $g^*(\alpha)$ is proper (TG, I, § 10). For every $t \in I$, the intersection of $A$ and the support of $g_t^*(\alpha)$ is then compact; the mapping $t \mapsto \int_A g_t^*(\alpha)$ is of class $C^1$ on $I$ and its derivative at the origin is given by the formula

cf. No. 8.6.