Divergence theorem

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In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem,[1][2] 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 over 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. In two dimensions, it is equivalent to Green's theorem.

The theorem is a special case of the more general Stokes' theorem.[3]

Intuition

If a fluid is flowing in some area, then the rate at which fluid flows out of a certain region within that area can be calculated by adding up the sources inside the region and subtracting 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.[4]

The divergence theorem is thus a conservation law which states that the volume total of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume's boundary.[5]

Mathematical statement

A region Template:Mvar bounded by the surface S = ∂V with the surface normal Template:Mvar
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 Template:Mvar is a subset of (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary Template:Mvar (also indicated with V = S). If F is a continuously differentiable vector field defined on a neighborhood of Template:Mvar, then we have:[6]

Template:Oiint

The left side is a volume integral over the volume Template:Mvar, the right side is the surface integral over the boundary of the volume Template:Mvar. The closed manifold V is quite generally the boundary of Template:Mvar 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 ndS.) The symbol within the two integrals stresses 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 Template:Mvar, 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 Template:Mvar and a vector field F, the result is
Template:Oiint
A special case of this is F = ∇ 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 F × G, the result is
Template:Oiint
  • Applying the divergence theorem to the product of a scalar function, f, and a non-zero constant vector c, the following theorem can be proven:[7]
Template:Oiint
  • Applying the divergence theorem to the cross-product of a vector field F and a non-zero constant vector c, the following theorem can be proven:[7]
Template:Oiint

Example

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

Suppose we wish to evaluate

Template:Oiint

where Template:Mvar is the unit sphere defined by

and F is the vector field

The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem, because the divergence theorem says that the integral is equal to:

where Template:Mvar is the unit ball:

Since the function Template:Mvar is positive in one hemisphere of Template:Mvar and negative in the other, in an equal and opposite way, its total integral over Template:Mvar is zero. The same is true for Template:Mvar:

Therefore,

Template:Oiint

because the unit ball Template:Mvar has volume {{ safesubst:#invoke:Unsubst||$B=4π/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

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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,[9] then later independently rediscovered by Gauss in 1813,[10] by Ostrogradsky, who also gave the first proof of the general theorem, in 1826,[11] by Green in 1828,[12] etc.[13] 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 Template:Mvar:

and the vector field:

The boundary of Template:Mvar is the unit circle, Template:Mvar, that can be represented parametrically by:

such that 0 ≤ s ≤ 2π where Template:Mvar units is the length arc from the point s = 0 to the point Template:Mvar on Template:Mvar. Then a vector equation of Template:Mvar is

At a point Template:Mvar on Template:Mvar:

Therefore,

Because M = 2y, {{ safesubst:#invoke:Unsubst||$B=M/x}} = 0, and because N = 5x, {{ safesubst:#invoke:Unsubst||$B=N/y}} = 0. Thus

Generalizations

Multiple dimensions

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

This equation is also known as the Divergence theorem.

When n = 2, this is equivalent to Green's theorem.

When n = 1, it reduces to the Fundamental theorem of calculus.

Tensor fields

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Writing the theorem in Einstein notation:

Template:Oiint

suggestively, replacing the vector field F with a rank-Template:Mvar tensor field Template:Mvar, this can be generalized to:[14]

Template:Oiint

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[15]).

Notes

  1. or less correctly as Gauss' theorem (see history for reason)
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  9. In his 1762 paper on sound, Lagrange treats a special case of the divergence theorem: Lagrange (1762) "Nouvelles recherches sur la nature et la propagation du son" (New researches on the nature and propagation of sound), Miscellanea Taurinensia (also known as: Mélanges de Turin ), 2: 11 - 172. This article is reprinted as: "Nouvelles recherches sur la nature et la propagation du son" in: J.A. Serret, ed., Oeuvres de Lagrange, (Paris, France: Gauthier-Villars, 1867), vol. 1, pages 151-316; on pages 263-265, Lagrange transforms triple integrals into double integrals using integration by parts.
  10. C. F. Gauss (1813) "Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata," Commentationes societatis regiae scientiarium Gottingensis recentiores, 2: 355-378; Gauss considered a special case of the theorem; see the 4th, 5th, and 6th pages of his article.
  11. Mikhail Ostragradsky presented his proof of the divergence theorem to the Paris Academy in 1826; however, his work was not published by the Academy. He returned to St. Petersburg, Russia, where in 1828-1829 he read the work that he'd done in France, to the St. Petersburg Academy, which published his work in abbreviated form in 1831.
    • His proof of the divergence theorem -- "Démonstration d'un théorème du calcul intégral" (Proof of a theorem in integral calculus) -- which he had read to the Paris Academy on February 13, 1826, was translated, in 1965, into Russian together with another article by him. See: Юшкевич А.П. (Yushkevich A.P.) and Антропова В.И. (Antropov V.I.) (1965) "Неопубликованные работы М.В. Остроградского" (Unpublished works of MV Ostrogradskii), Историко-математические исследования (Istoriko-Matematicheskie Issledovaniya / Historical-Mathematical Studies), 16: 49-96; see the section titled: "Остроградский М.В. Доказательство одной теоремы интегрального исчисления" (Ostrogradskii M. V. Dokazatelstvo odnoy teoremy integralnogo ischislenia / Ostragradsky M.V. Proof of a theorem in integral calculus).
    • M. Ostrogradsky (presented: November 5, 1828 ; published: 1831) "Première note sur la théorie de la chaleur" (First note on the theory of heat) Mémoires de l'Académie impériale des sciences de St. Pétersbourg, series 6, 1: 129-133; for an abbreviated version of his proof of the divergence theorem, see pages 130-131.
    • Victor J. Katz (May1979) "The history of Stokes' theorem," Mathematics Magazine, 52(3): 146-156; for Ostragradsky's proof of the divergence theorem, see pages 147-148.
  12. George Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Nottingham, England: T. Wheelhouse, 1838). A form of the "divergence theorem" appears on pages 10-12.
  13. Other early investigators who used some form of the divergence theorem include:
    • Poisson (presented: February 2, 1824 ; published: 1826) "Mémoire sur la théorie du magnétisme" (Memoir on the theory of magnetism), Mémoires de l'Académie des sciences de l'Institut de France, 5: 247-338; on pages 294-296, Poisson transforms a volume integral (which is used to evaluate a quantity Q) into a surface integral. To make this transformation, Poisson follows the same procedure that is used to prove the divergence theorem.
    • Frédéric Sarrus (1828) "Mémoire sur les oscillations des corps flottans" (Memoir on the oscillations of floating bodies), Annales de mathématiques pures et appliquées (Nismes), 19: 185-211.
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See also

External links

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