In vector calculus, the divergence theorem, also known as Gauss's theorem (Carl Friedrich Gauss), Ostrogradsky's theorem (Mikhail Vasilievich Ostrogradsky), or Gauss-Ostrogradsky theorem 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. Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... Johann Carl Friedrich Gauss (pronounced ,  ; in German usually Gauß, Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Mikhail Vasilievich Ostrogradsky (transcribed also Ostrogradskii, OstrogradskiÄ­, Mykhailo Vasylovych Ostrohradskyi[1]) (Михаил Васильевич Остроградский) (September 24, 1801 - January 1, 1862) was a Ukrainian mathematician, mechanician and physicist. ... flux in science and mathematics. ... Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ... An open surface with X-, Y-, and Z-contours shown. ...

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. flux in science and mathematics. ... For other uses, see Divergence (disambiguation). ...

The divergence theorem is an important result for the mathematics of engineering, in particular in electrostatics and fluid dynamics. Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... This box:      Paper shavings attracted by a charged CD Electrostatics is the branch of science that deals with the phenomena arising from what seems to be stationary electric charges. ... --68. ...

The theorem is a special case of the more general Stokes' theorem, which generalizes the fundamental theorem of calculus. Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ...

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.

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. In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

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 Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

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 n is the outward pointing unit normal field of the boundary ∂V. (dS may be used as a shorthand for ndS.) 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. In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain. ... In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

Note that the divergence theorem is a special case of the more general Stokes' theorem which generalizes the fundamental theorem of calculus. Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration. ...

Corollaries

By applying the divergence theorem in various contexts, other useful identities can be derived (cf. vector identities). This article lists a few helpful mathematical identities which are useful in vector algebra. ...

• 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 , in which case the theorem is the basis for Green's identities.

• Applying the divergence theorem to the cross-product of two vector fields , the result is

• Applying the divergence theorem to the product of a scalar function f and a nonzero constant vector, the following theorem can be proven:^[1]

• Applying the divergence theorem to the cross-product of a vector field F and a nonzero constant vector, the following theorem can be proven:^[1]

Greens identities are a set of three identities in vector calculus. ...

Example

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

Suppose we wish to evaluate Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

where S is the unit sphere defined by x² + y² + z² = 1 and F is the vector field In mathematics, unit ball and unit sphere refer to a ball with radius equal to 1. ... Vector field given by vectors of the form (−y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space. ...

The direct computation of this integral is quite difficult, but we can simplify it using the divergence theorem:

Since the functions y and z are odd on S (which is a symmetric set in respect to the coordinate planes), one has In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ...

Therefore,

because the unit ball W has volume 4π/3. For other uses, see Volume (disambiguation). ...

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. In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ... This box:      Paper shavings attracted by a charged CD Electrostatics is the branch of science that deals with the phenomena arising from what seems to be stationary electric charges. ...

Continuity equations

Main article: Continuity equation

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, 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). All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ... All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ... --68. ... This box:      Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

Inverse-square laws

Any "inverse-square law" can instead be written in a "Gauss's law"-type form (with a differential and integral form, as described above). Two examples are Gauss's law (in electrostatics), which follows from the inverse-square Coulomb's law, and Gauss's law for gravity, which follows from the inverse-square Newton's law of universal gravitation. The derivation of the Gauss's 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. In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ... This box:      Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows: The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each... Isaac Newtons theory of universal gravitation (part of classical mechanics) states the following: Every single point mass attracts every other point mass by a force pointing along the line combining the two. ...

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. A mathematical picture paints a thousand words: the Pythagorean theorem. ... Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ... 1762 was a common year starting on Friday (see link for calendar). ... Johann Carl Friedrich Gauss (pronounced ,  ; in German usually Gauß, Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Year 1813 (MDCCCXIII) was a common year starting on Friday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Wednesday of the 12-day slower Julian calendar). ... The title page to George Greens original essay on what is now known as Greens theorem. ... Year 1825 (MDCCCXXV) was a common year starting on Saturday (link will display the full calendar) of the Gregorian Calendar (or a common year starting on Thursday of the 12-day slower Julian calendar). ... Leopold I 1831 (MDCCCXXXI) was a common year starting on Saturday (see link for calendar). ... Mikhail Vasilievich Ostrogradsky (transcribed also Ostrogradskii, OstrogradskiÄ­, Mykhailo Vasylovych Ostrohradskyi[1]) (Михаил Васильевич Остроградский) (September 24, 1801 - January 1, 1862) was a Ukrainian mathematician, mechanician and physicist. ... In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special two-dimensional case of...

This article was originally based on the GFDL article from PlanetMath at http://planetmath.org/encyclopedia/Divergence.html GFDL redirects here. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

References

This article does not cite any references or sources. (May 2008) Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed.

External links

• Differential Operators and the Divergence Theorem at MathPages
• The Divergence (Gauss) Theorem by Nick Bykov, The Wolfram Demonstrations Project.
• Eric W. Weisstein, Divergence Theorem at MathWorld.