# Divergence theorem

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2010 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.

How to Cite This Entry:
Divergence theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergence_theorem&oldid=31341
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article