Divergence theorem

2020 Mathematics Subject Classification: Primary: 26B20 [MSN][ZBL]

The divergence theorem gives a formula in the integral calculus of functions in several variables that establishes a link between an $n$-fold integral over a domain and an $n-1$-fold integral over its boundary. The formula, which can be regarded as a direct generalization of the Fundamental theorem of calculus, is often referred to as: Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula or Gauss-Green-Ostrogradski formula.

Let us recall that, given an open set $U\subset \mathbb R^n$, a vector field on $U$ is a map $v: U \to \mathbb R^n$. If $v$ is differentiable and the components of the vector field are denoted by $v_1, \ldots, v_n$, then the divergence of $v$ is given by the function \[ {\rm div}\, v := \sum_{i=1}^n \frac{\partial v_i}{\partial x_i}\, . \]

The divergence theorem asserts that

Theorem 1 If $v$ is a $C^1$ vector field, $\partial U$ is regular (i.e. can be described locally as the graph of a $C^1$ function) and $U$ is bounded, then \begin{equation}\label{e:divergence_thm} \int_U {\rm div}\, v = \int_{\partial U} v\cdot \nu\, , \end{equation} where $\nu$ denotes the unit normal to $\partial U$ pointing towards the "exterior" (namely $\mathbb R^n \setminus \overline{U}$).

When the dimension $n$ is $1$ and $U$ is an interval $I =[a,b]$, the left hand side of \eqref{e:divergence_thm} is given by \[ \int_a^b f' (x)\, dx \] and the right hand side is given by $f(b)-f(a)$: the theorem is therefore a generalization of the Fundamental theorem of calculus. For larger $n$ the integral on the right hand side of \eqref{e:divergence_thm} is a surface integral, which is computed using the Area formula, and is called flux of the vector field $v$ through $\partial U$. If $v$ is compactly supported in a region $V$ where $U\cap V$ is the subgraph of a function $f$, then the flux of $v$ takes a simple form. More precisely, assume that $U\cap V = \{(x_1, \ldots, x_n)\in V : x_n < f(x_1, \ldots, x_{n-1})\}$. Then, under the assumption that $v$ vanishes outside $V$, we have \[ \int_{\partial U} v\cdot \nu = \int \left(v_n (x', f (x')) - \frac{\partial f}{\partial x_1} (x') v_1 (x', f(x')) - \ldots - \frac{\partial f}{\partial x_{n-1}} (x') v_{n-1} (x', f (x'))\right)\, dx'\, , \] where $x'= (x_1, \ldots, x_{n-1})$. This formula can be used, together with a partition of unity, to compute (or define) the flux of a general vector field $v$.

Remark 2 The three key assumptions of Theorem 1 can be all heavily relaxed:

• The boundedness of $U$ might be dropped if we assume that the vector field $v$ has suitable decay properties for $|x|\to \infty$.
• The regularity of $\partial U$ might be considerably weakened. For instance the theorem holds when $\partial U$ is piecewise $C^1$ and the singularities are "corner-like". More generaly, it still holds if $\partial U$ is Lipschitz. An important generalization holds for sets of finite perimeter: in this case the flux of the vector field through $\partial U$ must be suitably defined in a measure theoretic sense.
• The formula still holds when $v$ belongs to the Sobolev space $W^{1,p}$: in this case the right hand side of \eqref{e:divergence_thm} must be suitably interpreted, since $v$ is not necessarily continuous. Note that the almost everywhere differentiability of $v$ is not sufficient to guarantee \eqref{e:divergence_thm}, even when $v$ is continuous: see Absolute continuity for a counterexample.

Simultaneous weakenings of more than one assumption need to be handled with care.

Remark 3 The formula has also important generalizations of geometrical flavour. In particular, it holds on regular open subsets of Riemannian manifolds. A far-reaching generalization is given by the Stokes formula, using the language of differential forms.

Remark 4 Theorem 1 is attributed to different people. The $2$-dimensional case is credited often to Green, see [Gr]. The $3$-dimensional formula is attributed to Gauss, who proved a particular case in 1813, and to Ostrogradski (see [Os1]), who later generalized it to general dimension, [Os2]. Sometimes also Riemann is credited. However, it must be noted that the formula is already present in the works of Euler and other mathematicians of the 18th century.

References

[CH]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[Gr]	G. Green, "An essay on the application of mathematical analysis to the theories of electricity and magnetism" , Nottingham (1828) (Reprint: Mathematical papers, Chelsea, reprint, 1970, pp. 1–82) Zbl 21.0014.03
[Kr]	A.M. Krall, "Applied analysis" , Reidel (1986) pp. 380
[Os1]	M.V. Ostrogradski, Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles , 1 (1831) pp. 117–122
[Os2]	M.V. Ostrogradski, Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles , 1 (1838) pp. 35–58
[Sp]	M. Spivak, "Calculus on manifolds" , Benjamin (1965)
[Tr]	H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. Sect. 9.3.1
[Wi]	A.P. Wills, "Vector analysis with an introduction to tensor analysis" , Dover, reprint (1958) pp. 97ff
[vW ]	C. von Westenholz, "Differential forms in mathematical physics" , North-Holland (1981) pp. 286ff