Flux of a vector field

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

A concept in the integral calculus of functions in several variables. Let $\Omega\subset \mathbb R^n$ be an open set and $v$ a (continuous) vector field on $\Omega$, namely a map $v: \Omega\to \mathbb R^n$. If $\Sigma\subset\Omega$ is a $C^1$ $n-1$-dimensional surface oriented by a (continuous) unit normal $\nu$, the flux of the vector field $v$ through the surface $\Sigma$ is given by the integral \begin{equation}\label{e:flux} \int_\Sigma v\cdot \nu\, . \end{equation} The integral in \eqref{e:flux} is a surface integral, which is computed using the Area formula. If $\Sigma$ is given by the graph of a function $f: \mathbb R^{n-1} \supset V \to \mathbb R$ with its natural orientation, namely with \[ \nu (x_1, \ldots, x_{n-1}, f(x_1, \ldots, x_{n-1})) = \frac{(-\nabla f, 1)}{\sqrt{1+|\nabla f|^2}} (x_1, \ldots, x_{n-1})\, , \] then we have the useful formula \[ \int_{\partial U} v\cdot \nu = \int \left(v_n (x', f (x')) - \frac{\partial f}{\partial x_1} (x') v_1 (x', f(x')) - \ldots - \frac{\partial f}{\partial x_{n-1}} (x') v_{n-1} (x', f (x'))\right)\, dx'\, , \] where $x' = (x_1, \ldots , x_{n-1})$. The latter formula can be used to define the flux of a vector field over a general $C^1$ surface using a partition of unity.

An alternative powerful way to define the surface integral in \eqref{e:flux} is to resort to differential forms and their integration of manifolds, see [Sp]. More precisely, if $v_1, \ldots, v_n$ are the components of the vector function $v$, it is convenient to introduce the $n-1$-form \[ \omega = \sum_{i=1}^n (-1)^{i-1} v_i dx_1 \wedge \ldots \wedge dx_{i-1}\wedge dx_{i+1} \wedge \ldots \wedge dx_n\, . \] Then it turns out that the integral in \eqref{e:flux} is in fact \[ \int_{\Sigma} \omega\, . \]

The divergence theorem relates the flux of a differentiable vector field of $v$ through the boundary of a regular open set $U$ to the integral over $U$ of the divergence of $v$. This important theorem (which goes also under the name Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula or Gauss-Green-Ostrogradski formula) is a generalization of the Fundamental theorem of calculus and it is a particular case of the more general Stokes theorem on integral of differential forms.

References

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[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
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