Divergence theorem
Encyclopedia
In vector calculus, the divergence theorem, also known as Gauss' 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.
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
.
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.
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
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.
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
), Ostrogradsky's theorem (Mikhail Vasilievich Ostrogradsky
Mikhail Vasilievich Ostrogradsky
Mikhail Vasilyevich Ostrogradsky was an Russian / Ukrainian mathematician, mechanician and physicist...
), or Gauss–Ostrogradsky theorem is a result that relates the flow (that is, flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
) of a vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
through a surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
to the behavior of the vector field inside the surface.
More precisely, the divergence theorem states that the outward flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
of a vector field through a closed surface is equal to the volume integral
Volume integral
In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain....
of the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
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
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
, in particular in electrostatics
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
and fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
.
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
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
.
The theorem is a special case of the more general Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
.
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
Conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
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.
Mathematical statement
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
and has a piecewise
Piecewise
On mathematics, a piecewise-defined function is a function whose definition changes depending on the value of the independent variable...
smooth
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
boundary
Boundary (topology)
In topology and mathematics in general, 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 precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have
The left side is a volume integral
Volume integral
In mathematics — in particular, in multivariable calculus — a volume integral refers to an integral over a 3-dimensional domain....
over the volume V, the right side is the surface integral
Surface integral
In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral...
over the boundary of the volume V. The closed manifold ∂V is quite generally the boundary of V oriented by outward-pointing normals
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
, 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).- 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 identitiesGreen's identitiesIn mathematics, Green's identities are a set of three identities in vector calculus. They are named after the mathematician George Green, who discovered Green's theorem.-Green's first identity:...
.
- 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 non-zero constant vector, the following theorem can be proven:
- 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:
Example
Suppose we wish to evaluate
where S is the unit sphereUnit sphereIn mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
defined by
and F is the vector fieldVector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem:
Since the functions
and 
are odd on 
(which is a symmetric set with respect to the coordinate planes), one has
Therefore,
because the unit sphere W has volumeVolumeVolume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....
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 lawGauss's lawIn physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
(in electrostaticsElectrostaticsElectrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
), Gauss's law for magnetism, and Gauss's law for gravity.
Continuity equations
Continuity equationContinuity equationA continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
s offer more examples of laws with both differential and integral forms, related to each other by the divergence theorem. In fluid dynamicsFluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, electromagnetismElectromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
, quantum mechanicsQuantum mechanicsQuantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, relativity theory, and a number of other fields, there are continuity equationContinuity equationA continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
s 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).
Inverse-square laws
Any inverse-square lawInverse-square lawIn physics, an inverse-square law is any physical law stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity....
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 lawCoulomb's lawCoulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
, and Gauss' law for gravityGauss' law for gravityIn physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics which is essentially equivalent to Newton's law of universal gravitation...
, which follows from the inverse-square Newton's law of universal gravitationNewton's law of universal gravitationNewton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...
. 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.
History
The theoremTheoremIn mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
was first discovered by Joseph Louis LagrangeJoseph Louis LagrangeJoseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...
in 1762, then later independently rediscovered by Carl Friedrich GaussCarl Friedrich GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
in 1813, by George GreenGeorge GreenGeorge Green was a British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism...
in 1825 and in 1831 by Mikhail Vasilievich OstrogradskyMikhail Vasilievich OstrogradskyMikhail Vasilyevich Ostrogradsky was an Russian / Ukrainian mathematician, mechanician and physicist...
, who also gave the first proof of the theorem. Subsequently, variations on the divergence theorem are called Gauss's theorem, Green's theoremGreen's theoremIn mathematics, Green's 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...
, and Ostrogradsky's theorem.
Examples
Verify the planar variant of the divergence theorem for a region R, with F(x,y) = 2yi + 5xj, where R is the region bounded by the circle
Solution: The boundary of R is the unit circle, C, that can be represented parametrically by:
such that
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
At a point
on C, 
. Therefore,
Because
, 
, and because 
, 
. Thus
External links
- Differential Operators and the Divergence Theorem at MathPages
- The Divergence (Gauss) Theorem by Nick Bykov, Wolfram Demonstrations ProjectWolfram Demonstrations ProjectThe Wolfram Demonstrations Project is hosted by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...
.
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