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Divergence theorem

In vector calculus, the divergence theorem, also known as Ostrogradsky's theorem,[1] is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface.

More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence of 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 engineering, in particular in electrostatics and fluid dynamics.

In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus.

The theorem is a special case of the more general Stokes' theorem.[2]

Contents


Intuition

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.[3]

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.[4]

Mathematical statement

A region V bounded by the surface S= V with the surface normal n The divergence theorem can be used to calculate a flux through a closed surface that fully encloses a volume, like any of the surfaces on the left. It can not directly be used to calculate the flux through surfaces with boundaries, like those on the right. (Surfaces are blue, boundaries are red.)

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 S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have[5]

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. The closed manifold V is quite generally the boundary of V oriented by outward-pointing normals, and n is the outward pointing unit normal field of the boundary V. (dS may be used as a shorthand for n dS.) By the symbol within the two integrals it is stressed once more that V is a closed surface. 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.

Corollaries

By applying the divergence theorem in various contexts, other useful identities can be derived (cf. vector identities).[6]

  • Applying the divergence theorem to the product of a scalar function g and a vector field F, the result is
A special case of this is \scriptstyle \mathbf{F}=\nabla f, in which case the theorem is the basis for Green's identities.
  • Applying the divergence theorem to the cross-product of two vector fields \scriptstyle \mathbf{F}\times \mathbf{G}, the result is
  • Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector, the following theorem can be proven:[7]
  • Applying the divergence theorem to the cross-product of a vector field F and a non-zero constant vector, the following theorem can be proven:[7]

Example

The vector field corresponding to the example shown. Note, vectors may point into or out of the sphere.

Suppose we wish to evaluate

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 we can simplify the derivation of the result using the divergence theorem:

|- valign="top" | ||   || \begin{align} &= \int\!\!\!\!\int\!\!\!\!\int_W\left(\nabla\cdot\mathbf{F}\right) \, dV\\ &= 2\int\!\!\!\!\int\!\!\!\!\int_W\left(1+y+z\right) \, dV\\ &= 2\int\!\!\!\!\int\!\!\!\!\int_W \,dV + 2\int\!\!\!\!\int\!\!\!\!\int_W y \,dV + 2\int\!\!\!\!\int\!\!\!\!\int_W z \,dV. \end{align} |}

where W is the unit ball (i.e., the interior of the unit sphere, \scriptstyle x^2+y^2+z^2\leq 1). Since the function \scriptstyle y is positive in one hemisphere of W and negative in the other, in an equal and opposite way, its total integral over W is zero. The same is true for \scriptstyle z:

\int\!\!\!\int\!\!\!\int_W y\, dV = \int\!\!\!\int\!\!\!\int_W z\, dV = 0.

Therefore,

because the unit ball W has volume \scriptstyle \frac{4\pi}{3}.

Applications

Differential form and integral form of physical laws

As a result of the divergence theorem, a host of physical laws can be written in both a differential form (where one quantity is the divergence of another) and an integral form (where the flux of one quantity through a closed surface is equal to another quantity). Three examples are Gauss's law (in electrostatics), Gauss's law for magnetism, and Gauss's law for gravity.

Continuity equations

Continuity equations offer more examples of laws with both differential and integral forms, related to each other by the divergence theorem. In fluid dynamics, electromagnetism, quantum mechanics, relativity theory, and a number of other fields, there are continuity equations that describe the conservation of mass, momentum, energy, probability, or other quantities. Generically, these equations state that the divergence of the flow of the conserved quantity is equal to the distribution of sources or sinks of that quantity. The divergence theorem states that any such continuity equation can be written in a differential form (in terms of a divergence) and an integral form (in terms of a flux).[8]

Inverse-square laws

Any inverse-square law can instead be written in a Gauss' law-type form (with a differential and integral form, as described above). Two examples are Gauss' law (in electrostatics), which follows from the inverse-square Coulomb's law, and Gauss' law for gravity, which follows from the inverse-square Newton's law of universal gravitation. The derivation of the Gauss' law-type equation from the inverse-square formulation (or vice-versa) is exactly the same in both cases; see either of those articles for details.[8]

History

The theorem was first discovered by Lagrange in 1762, then later independently rediscovered by Gauss in 1813, by Green in 1825 and in 1831 by Ostrogradsky, who also gave the first proof of the theorem. Subsequently, variations on the divergence theorem are correctly called Ostrogradsky's theorem, but also commonly Gauss's theorem, or Green's theorem.

Examples

To verify the planar variant of the divergence theorem for a region R, where

\mathbf{F}(x,y)=2 y\mathbf{i} + 5x \mathbf{j} ,\

and R is the region bounded by the circle

x^2 + y^2 = 1.\

The boundary of R is the unit circle, C, that can be represented parametrically by:

x\, = \cos(s);\ y\, = \sin(s)\

such that \scriptstyle 0\, \le \,s\, \le \,2\pi where s units is the length arc from the point s = 0 to the point P on C. Then a vector equation of C is

\mathbf{C}(s) = \cos(s)\mathbf{i} + \sin(s)\mathbf{j}.\

At a point P on C:

\mathbf{P}\, = \,(\cos(s),\, \sin(s)) \, \Rightarrow \, \mathbf{F}\, = \,2\sin(s)\mathbf{i} + 5\cos(s)\mathbf{j} \, .

Therefore,

\begin{align}\oint_C \mathbf{F} \cdot \mathbf{N}\, ds &= \,\int_{0}^{2 \pi} ( 2 \sin s \mathbf{i} + 5 \cos s \mathbf{j}) \cdot (\cos s \mathbf{i} + \sin s \mathbf{j})\, ds\\ &= \,\int_{0}^{2 \pi} (2 \sin s \cos s + 5 \sin s \cos s)\, ds\\ &= \,7\int_{0}^{2 \pi} \sin s \cos s\, ds\\ &= \,0.\end{align}\

Because \scriptstyle M\, = \,2y\,, \scriptstyle \frac{\partial M}{\partial x} = \,0, and because \scriptstyle N\, = \,5x\,, \scriptstyle \frac{\partial N}{\partial y} = \,0. Thus

\iint\limits_R \, \operatorname{div} \mathbf{F} \,dA = \iint\limits_R \left (\frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} \right) \, dA = 0.

Generalizations

Multiple dimensions

One can use the general Stokes' Theorem to equate the n-dimensional volume integral of the divergence of a vector field F over a region U to the (n-1)-dimensional surface integral of F over the boundary of U:

\int_U \nabla \cdot \mathbf{F} \, dV_n = \oint_{\partial U} \mathbf{F} \cdot \mathbf{n} \, dS_{n-1}

This equation is also known as the Divergence Theorem.

Tensor fields

Writing the theorem in index notation:

suggestively, replacing the vector field F with a rank-n tensor field T, this can be generalized to:[9]

where on each side, tensor contraction occurs for at least one index. This form of the theorem is still in 3d, each index takes values 1, 2, and 3. It can be generalized further still to higher (or lower) dimensions (for example to 4d spacetime in general relativity[10]).

Notes

External links


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

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