Divergence theorem

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In vector calculus, the divergence theorem, also known as Gauss' theorem, Ostrogradsky's theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface.

More precisely, the divergence theorem states that the outward flux of a vector field through a surface is equal to the triple integral of the divergence on the region inside the surface. Intuitively, it states that the sum of all sources minus the sum of all sinks gives the net flow out of a region.

The divergence theorem is an important result for the mathematics of physics, in particular in electrostatics and fluid dynamics.

1 Intuition
2 Mathematical statement
3 Example
4 Applications
5 History

[edit] Intuition

The intuitive content is simple. If a fluid is flowing in some area, and we wish to know how much fluid flows out of a certain region within that area, then we need to add up the sources inside the region and subtract the sinks. The fluid flow is represented by a vector field, and the vector field's divergence at a given point describes the strength of the source or sink there. So, integrating the field's divergence over the interior of the region should equal the integral of the vector field over the region's boundary. The divergence theorem says that this is true.

The divergence theorem is thus a conservation law which states that the volume total of all sinks and sources, the volume integral of the divergence, is equal to the net flow across the volume's boundary.

[edit] Mathematical statement

A region V bounded by the surface S=∂V with the surface normal n.

Suppose V is a subset of Rⁿ (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have

$\iiint\limits_V\left(\nabla\cdot\mathbf{F}\right)dV=\iint\limits_{\part V} \mathbf{F} \cdot \mathbf{n}\, dS.$

The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. Here ∂V is quite generally the boundary of V oriented by outward-pointing normals, and dS is shorthand for ndS, the outward pointing unit normal field of the boundary ∂V. In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume V, and the right-hand side represents the total flow across the boundary ∂V.

The divergence theorem is frequently applied in these following variants (cf. vector identities):

$\iiint\limits_V\mathbf{F}\cdot \left(\nabla g\right) + g \left(\nabla\cdot \mathbf{F}\right)dV=\iint\limits_{\part V}g \mathbf{F}\cdot d\mathbf{S}$

(this is the basis for Green's identities, if $\mathbf{F}=\nabla f$ ),

$\iiint\limits_V \nabla g dV=\iint\limits_{\part V} g d\mathbf{S},$

$\iiint\limits_V \mathbf{G}\cdot\left(\nabla\times\mathbf{F}\right) - \mathbf{F}\cdot \left( \nabla\times\mathbf{G}\right) dV = \iint\limits_{\part V}\left(\mathbf{F}\times\mathbf{G}\right)\cdot d\mathbf{S},$

$\iiint\limits_V \nabla\times\mathbf{F} dV = \iint\limits_{\part V}d\mathbf{S} \times\mathbf{F}.$

Note that the divergence theorem is a special case of the more general Stokes theorem which generalizes the fundamental theorem of calculus.

[edit] Example

A vector field on a sphere (this is not the field in the example).

Suppose we wish to evaluate

$\iint\limits_S\mathbf{F}\cdot \mathbf{n}dS,$

where S is the unit sphere defined by $x 2 + y 2 + z 2 = 1$ and F is the vector field

$\mathbf{F} = 2 x\mathbf{i}+y^2\mathbf{j}+z^2\mathbf{k}.$

The direct computation of this integral is quite difficult, but can be simplified using the divergence theorem:

$\iint\limits_S\mathbf{F}\cdot \mathbf{n} dS=\iiint\limits_W\left(\nabla\cdot\mathbf{F}\right)dV=2\iiint\limits_W\left(1+y+z\right)dV$

$= 2\iiint\limits_W dV + 2\iiint\limits_W y dV + 2\iiint\limits_W z dV.$

Since the functions $y$ and $z$ are odd on $S$ (which is a symmetric set in respect to the coordinate planes), one has

$\iiint\limits_W y dV = \iiint\limits_W z dV = 0.$

Therefore,

$2\iiint\limits_W\left(1+y+z\right)dV = 2\iiint\limits_W dV = \frac{8\pi}{3}$

because the unit ball W has volume 4π/3.

[edit] Applications

[edit] Fluid dynamics

Used to simplify the conservation of mass, momentum and energy equations.

[edit] Electrostatics

Applied to an electrostatic field we get Gauss's law: the divergence is a constant times the volume charge density.

[edit] Gravity

Applied to a gravitational field we get that the surface integral is -4πG times the mass inside, regardless of how the mass is distributed, and regardless of any masses outside.

[edit] Spherically symmetric mass distribution

In the case of a spherically symmetric mass distribution we can conclude from this that the field strength at a distance r from the center is inward with a magnitude of G/r² times only the total mass within a smaller distance than r. All the mass at a greater distance than r from the center can be ignored.

For example, a hollow sphere does not produce any gravity inside. The gravitational field inside is the same as if the hollow sphere were not there (i.e. the field is that of any masses inside and outside the sphere only).

[edit] Cylindrically symmetric mass distribution

In the case of an infinite cylindrically symmetric mass distribution we can conclude that the field strength at a distance r from the center is inward with a magnitude of 2G/r times the total mass per unit length at a smaller distance, regardless of any masses at a larger distance.

For example, an infinite hollow cylinder does not produce any gravity inside.

[edit] Bouguer plate

We can conclude that for an infinite, flat plate (Bouguer plate) of thickness H gravity outside the plate is perpendicular to the plate, towards it, with magnitude 2πG times the mass per unit area, independent of the distance to the plate (see also gravity anomalies).

More generally, for a mass distribution with the density depending on one Cartesian coordinate z only, gravity for any z is 2πG times the difference in mass per unit area on either side of this z value.

In particular, a combination of two equal parallel infinite plates does not produce any gravity inside.

[edit] History

The theorem was first discovered by Joseph Louis Lagrange in 1762, then later independently rediscovered by Carl Friedrich Gauss in 1813, by George Green in 1825 and in 1831 by Mikhail Vasilievich Ostrogradsky, who also gave the first proof of the theorem. Subsequently, variations on the Divergence theorem are called Gauss's Theorem, Green's theorem, and Ostrogradsky's theorem.

This article was originally based on the GFDL article from PlanetMath at http://planetmath.org/encyclopedia/Divergence.html

Retrieved from "http://en.wikipedia.org/wiki/Divergence_theorem"

Categories: Vector calculus | Mathematical theorems