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In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the '' flux'' of a vector field through a closed surface to the ''
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
'' of the field in the volume enclosed. More precisely, the divergence theorem states that the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
of a vector field over a closed surface, which is called the "flux" through the 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 of the field in a region (with sinks regarded as negative sources) gives the net flux out of the region". The divergence theorem is an important result for the mathematics of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, particularly in electrostatics and
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
. In these fields, it is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ...
. In two dimensions, it is equivalent to
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively orie ...
.


Explanation using liquid flow

Vector fields are often illustrated using the example of the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
field of a fluid, such as a gas or liquid. A moving liquid has a velocity—a speed and a direction—at each point, which can be represented by a vector, so that the velocity of the liquid at any moment forms a vector field. Consider an imaginary closed surface ''S'' inside a body of liquid, enclosing a volume of liquid. The flux of liquid out of the volume is equal to the volume rate of fluid crossing this surface, i.e., the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
of the velocity over the surface. Since liquids are incompressible, the amount of liquid inside a closed volume is constant; if there are no sources or sinks inside the volume then the flux of liquid out of ''S'' is zero. If the liquid is moving, it may flow into the volume at some points on the surface ''S'' and out of the volume at other points, but the amounts flowing in and out at any moment are equal, so the ''net'' flux of liquid out of the volume is zero. However if a ''source'' of liquid is inside the closed surface, such as a pipe through which liquid is introduced, the additional liquid will exert pressure on the surrounding liquid, causing an outward flow in all directions. This will cause a net outward flow through the surface ''S''. The flux outward through ''S'' equals the volume rate of flow of fluid into ''S'' from the pipe. Similarly if there is a ''sink'' or drain inside ''S'', such as a pipe which drains the liquid off, the external pressure of the liquid will cause a velocity throughout the liquid directed inward toward the location of the drain. The volume rate of flow of liquid inward through the surface ''S'' equals the rate of liquid removed by the sink. If there are multiple sources and sinks of liquid inside ''S'', the flux through the surface can be calculated by adding up the volume rate of liquid added by the sources and subtracting the rate of liquid drained off by the sinks. The volume rate of flow of liquid through a source or sink (with the flow through a sink given a negative sign) is equal to the ''
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
'' of the velocity field at the pipe mouth, so adding up (integrating) the divergence of the liquid throughout the volume enclosed by ''S'' equals the volume rate of flux through ''S''. This is the divergence theorem. The divergence theorem is employed in any conservation law which states that the total volume of all sinks and sources, that is the volume integral of the divergence, is equal to the net flow across the volume's boundary.


Mathematical statement

Suppose is a subset of \mathbb^n (in the case of represents a volume in
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
) which is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
and has a
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
smooth boundary (also indicated with \partial V = S). If is a continuously differentiable vector field defined on a neighborhood of , then: : The left side is a volume integral over the volume , the right side is the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one ...
over the boundary of the volume . The closed manifold \partial V is oriented by outward-pointing normals, and \mathbf is the outward pointing unit normal at each point on the boundary \partial V. (\mathrm \mathbf may be used as a shorthand for \mathbf \mathrm S.) In terms of the intuitive description above, the left-hand side of the equation represents the total of the sources in the volume , and the right-hand side represents the total flow across the boundary .


Informal derivation

The divergence theorem follows from the fact that if a volume is partitioned into separate parts, the flux out of the original volume is equal to the sum of the flux out of each component volume. This is true despite the fact that the new subvolumes have surfaces that were not part of the original volume's surface, because these surfaces are just partitions between two of the subvolumes and the flux through them just passes from one volume to the other and so cancels out when the flux out of the subvolumes is summed. See the diagram. A closed, bounded volume is divided into two volumes and by a surface ''(green)''. The flux out of each component region is equal to the sum of the flux through its two faces, so the sum of the flux out of the two parts is :\Phi(V_\text) + \Phi(V_\text) = \Phi_\text + \Phi_\text + \Phi_\text + \Phi_\text where and are the flux out of surfaces and , is the flux through out of volume 1, and is the flux through out of volume 2. The point is that surface is part of the surface of both volumes. The "outward" direction of the normal vector \mathbf is opposite for each volume, so the flux out of one through is equal to the negative of the flux out of the other :\Phi_\text = \iint_ \mathbf \cdot \mathbf \; \mathrmS = -\iint_ \mathbf \cdot (-\mathbf) \; \mathrmS = -\Phi_\text so these two fluxes cancel in the sum. Therefore :\Phi(V_\text) + \Phi(V_\text) = \Phi_\text + \Phi_\text Since the union of surfaces and is :\Phi(V_\text) + \Phi(V_\text) = \Phi(V)
This principle applies to a volume divided into any number of parts, as shown in the diagram. Since the integral over each internal partition ''(green surfaces)'' appears with opposite signs in the flux of the two adjacent volumes they cancel out, and the only contribution to the flux is the integral over the external surfaces ''(grey)''. Since the external surfaces of all the component volumes equal the original surface. :\Phi(V) = \sum_ \Phi(V_\text)
The flux out of each volume is the surface integral of the vector field over the surface :\iint_ \mathbf \cdot \mathbf \; \mathrmS = \sum_ \iint_ \mathbf \cdot \mathbf \; \mathrmS The goal is to divide the original volume into infinitely many infinitesimal volumes. As the volume is divided into smaller and smaller parts, the surface integral on the right, the flux out of each subvolume, approaches zero because the surface area approaches zero. However, from the definition of
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
, the ratio of flux to volume, \frac = \frac \iint_ \mathbf \cdot \mathbf \; \mathrmS, the part in parentheses below, does not in general vanish but approaches the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
as the volume approaches zero. :\iint_ \mathbf \cdot \mathbf \; \mathrmS = \sum_ \left(\frac \iint_ \mathbf \cdot \mathbf \; \mathrmS\right) , V_\text, As long as the vector field has continuous derivatives, the sum above holds even in the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
when the volume is divided into infinitely small increments :\iint_ \mathbf \cdot \mathbf \; \mathrmS = \lim_\sum_ \left(\frac\iint_ \mathbf \cdot \mathbf \; \mathrmS\right) , V_\text, As , V_\text, approaches zero volume, it becomes the infinitesimal , the part in parentheses becomes the divergence, and the sum becomes a volume integral over Since this derivation is coordinate free, it shows that the divergence does not depend on the coordinates used.


Proofs


For bounded open subsets of Euclidean space

We are going to prove the following: Proof of Theorem. (1) The first step is to reduce to the case where u \in C_c^1(\mathbb^n). Pick \phi \in C_c^(O) such that \phi = 1 on \overline. Note that \phi u \in C_c^(O) \subset C_c^1(\mathbb^n) and \phi u = u on \overline. Hence it suffices to prove the theorem for \phi u. Hence we may assume that u \in C_c^1(\mathbb^n). (2) Let x_0 \in \partial \Omega be arbitrary. The assumption that \overline has C^1 boundary means that there is an open neighborhood U of x_0 in \mathbb^n such that \partial \Omega \cap U is the graph of a C^1 function with \Omega \cap U lying on one side of this graph. More precisely, this means that after a translation and rotation of \Omega, there are r > 0 and h > 0 and a C^1 function g : \mathbb^ \to \mathbb, such that with the notation x' = (x_1, \dots, x_), it holds that U = \ and for x \in U, \begin x_n = g(x') & \implies x \in \partial \Omega, \\ -h < x_n - g(x') < 0 & \implies x \in \Omega, \\ 0 < x_n - g(x') < h & \implies x \notin \Omega. \\ \end Since \partial \Omega is compact, we can cover \partial \Omega with finitely many neighborhoods U_1, \dots, U_N of the above form. Note that \ is an open cover of \overline = \Omega \cup \partial \Omega. By using a C^ partition of unity subordinate to this cover, it suffices to prove the theorem in the case where either u has compact
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
in \Omega or u has compact support in some U_j. If u has compact support in \Omega, then for all i \in \, \int_ u_\,dV = \int_u_\,dV = \int_ \int_^u_(x)\,dx_i\,dx' = 0 by the fundamental theorem of calculus, and \int_u\nu_i\,dS = 0 since u vanishes on a neighborhood of \partial \Omega. Thus the theorem holds for u with compact support in \Omega. Thus we have reduced to the case where u has compact support in some U_j. (3) So assume u has compact support in some U_j. The last step now is to show that the theorem is true by direct computation. Change notation to U = U_j, and bring in the notation from (2) used to describe U. Note that this means that we have rotated and translated \Omega. This is a valid reduction since the theorem is invariant under rotations and translations of coordinates. Since u(x) = 0 for , x', \geq r and for , x_n - g(x'), \geq h, we have for each i \in \ that \begin \int_u_\,dV &= \int_\int_^u_(x', x_n)\,dx_n\,dx' \\ &= \int_\int_^u_(x', x_n)\,dx_n\,dx'. \end For i = n we have by the fundamental theorem of calculus that \int_\int_^u_(x', x_n)\,dx_n\,dx' = \int_u(x', g(x'))\,dx'. Now fix i \in \. Note that \int_\int_^u_(x', x_n)\,dx_n\,dx' = \int_\int_^u_(x', g(x') + s)\,ds\,dx' Define v : \mathbb^ \to \mathbb by v(x', s) = u(x', g(x') + s). By the chain rule, v_(x', s) = u_(x', g(x') + s) + u_(x', g(x') + s)g_(x'). But since v has compact support, we can integrate out dx_i first to deduce that \int_\int_^v_(x', s)\,ds\,dx' = 0. Thus \begin \int_\int_^u_(x', g(x') + s)\,ds\,dx' &= \int_\int_^-u_(x', g(x') + s)g_(x')\,ds\,dx' \\ &= \int_-u(x', g(x'))g_(x')\,dx'. \end In summary, with \nabla u = (u_, \dots, u_) we have \int_\nabla u\,dV = \int_\int_^\nabla u\,dV = \int_u(x', g(x'))(-\nabla g(x'), 1)\,dx'. Recall that the outward unit normal to the graph \Gamma of g at a point (x', g(x')) \in \Gamma is \nu(x', g(x')) = \frac(-\nabla g(x'), 1) and that the surface element dS is given by dS = \sqrt\,dx'. Thus \int_\nabla u\,dV = \int_u\nu\,dS. This completes the proof.


For compact Riemannian manifolds with boundary

We are going to prove the following: Proof of Theorem. We use the Einstein summation convention. By using a partition of unity, we may assume that u and X have compact support in a coordinate patch O \subset \overline. First consider the case where the patch is disjoint from \partial \Omega. Then O is identified with an open subset of \mathbb^n and integration by parts produces no boundary terms: \begin (\operatorname u, X) &= \int_\langle \operatorname u, X \rangle \sqrt\,dx \\ &= \int_\partial_j u X^j \sqrt\,dx \\ &= -\int_u \partial_j(\sqrtX^j)\,dx \\ &= -\int_ u \frac\partial_j(\sqrtX^j)\sqrt\,dx \\ &= (u, -\frac\partial_j(\sqrtX^j)) \\ &= (u, -\operatorname X). \end In the last equality we used the Voss-Weyl coordinate formula for the divergence, although the preceding identity could be used to define -\operatorname as the formal adjoint of \operatorname. Now suppose O intersects \partial \Omega. Then O is identified with an open set in \mathbb_^n = \. We zero extend u and X to \mathbb_+^n and perform integration by parts to obtain \begin (\operatorname u, X) &= \int_\langle \operatorname u, X \rangle \sqrt\,dx \\ &= \int_\partial_j u X^j \sqrt\,dx \\ &= (u, -\operatorname X) - \int_u(x', 0)X^n(x', 0)\sqrt\,dx', \end where dx' = dx_1 \dots dx_. By a variant of the straightening theorem for vector fields, we may choose O so that \frac is the inward unit normal -N at \partial \Omega. In this case \sqrt\,dx' = \sqrt\,dx' = dS is the volume element on \partial \Omega and the above formula reads (\operatorname u, X) = (u, -\operatorname X) + \int_u\langle X, N \rangle \,dS. This completes the proof.


Corollaries

By replacing in the divergence theorem with specific forms, other useful identities can be derived (cf. vector identities). * With \mathbf\rightarrow \mathbfg for a scalar function and a vector field , :: :A special case of this is \mathbf = \nabla f, in which case the theorem is the basis for
Green's identities In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's ...
. * With \mathbf\rightarrow \mathbf\times \mathbf for two vector fields and , where \times denotes a cross product, :: * With \mathbf\rightarrow \mathbf\cdot \mathbf for two vector fields and , where \cdot denotes a dot product, :: * With \mathbf\rightarrow f\mathbf for a scalar function and vector field c:MathWorld
/ref> :: :The last term on the right vanishes for constant \mathbf or any divergence free (solenoidal) vector field, e.g. Incompressible flows without sources or sinks such as phase change or chemical reactions etc. In particular, taking \mathbf to be constant: :: * With \mathbf\rightarrow \mathbf\times\mathbf for vector field and constant vector c: :: : By reordering the
triple product In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector- ...
on the right hand side and taking out the constant vector of the integral, :: : Hence, ::


Example

Suppose we wish to evaluate : where is the unit sphere defined by :S = \left \, and is the vector field :\mathbf = 2x\mathbf+y^2\mathbf+z^2\mathbf. 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: :\iiint_W (\nabla \cdot \mathbf)\,\mathrmV = 2\iiint_W (1 + y + z)\, \mathrmV = 2\iiint_W \mathrmV + 2\iiint_W y\, \mathrmV + 2\iiint_W z\, \mathrmV, where is the unit ball: :W = \left \. Since the function is positive in one hemisphere of and negative in the other, in an equal and opposite way, its total integral over is zero. The same is true for : :\iiint_W y\, \mathrmV = \iiint_W z\, \mathrmV = 0. Therefore, : because the unit ball has
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
.


Applications


Differential and integral forms 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 physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it st ...
(in electrostatics), Gauss's law for magnetism, and
Gauss's law for gravity In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux (surface int ...
.


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 In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
,
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
relativity theory The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena ...
, 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).


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 physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it st ...
(in electrostatics), which follows from the inverse-square
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
, and
Gauss's law for gravity In physics, Gauss's law for gravity, also known as Gauss's flux theorem for gravity, is a law of physics that is equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. It states that the flux (surface int ...
, 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.


History

Joseph-Louis Lagrange introduced the notion of surface integrals in 1760 and again in more general terms in 1811, in the second edition of his '' Mécanique Analytique.'' Lagrange employed surface integrals in his work on fluid mechanics. He discovered the divergence theorem in 1762. Carl Friedrich Gauss was also using surface integrals while working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem. He proved additional special cases in 1833 and 1839. But it was
Mikhail Ostrogradsky Mikhail Vasilyevich Ostrogradsky (transcribed also ''Ostrogradskiy'', Ostrogradskiĭ) (russian: Михаи́л Васи́льевич Острогра́дский, ua, Миха́йло Васи́льович Острогра́дський; 24 Sep ...
, who gave the first proof of the general theorem, in 1826, as part of his investigation of heat flow. Special cases were proven by George Green in 1828 in ''An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism'', Siméon Denis Poisson in 1824 in a paper on elasticity, and Frédéric Sarrus in 1828 in his work on floating bodies.


Worked examples


Example 1

To verify the planar variant of the divergence theorem for a region R: :R = \left \, and the vector field: : \mathbf(x,y)= 2 y\mathbf + 5x \mathbf. The boundary of R is the unit circle, C, that can be represented parametrically by: :x = \cos(s), \quad y = \sin(s) such that 0 \leq s \leq 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 :C(s) = \cos(s)\mathbf + \sin(s)\mathbf. At a point P on C: : P = (\cos(s),\, \sin(s)) \, \Rightarrow \, \mathbf = 2\sin(s)\mathbf + 5\cos(s)\mathbf. Therefore, :\begin \oint_C \mathbf \cdot \mathbf\, \mathrms &= \int_0^ (2 \sin(s) \mathbf + 5 \cos(s) \mathbf) \cdot (\cos(s) \mathbf + \sin(s) \mathbf)\, \mathrms\\ &= \int_0^ (2 \sin(s) \cos(s) + 5 \sin(s) \cos(s))\, \mathrms\\ &= 7\int_0^ \sin(s) \cos(s)\, \mathrms\\ &= 0. \end Because M = 2y, we can evaluate and because \frac = 0. Thus :\iint_R \, \mathbf\cdot\mathbf \, \mathrmA = \iint_R \left (\frac + \frac \right) \, \mathrmA = 0.


Example 2

Let's say we wanted to evaluate the flux of the following vector field defined by \mathbf=2x^2 \textbf +2y^2 \textbf +2z^2\textbf bounded by the following inequalities: :\left\ \left\ \left\ By the divergence theorem, : We now need to determine the divergence of \textbf. If \mathbf is a three-dimensional vector field, then the divergence of \textbf is given by \nabla \cdot \textbf = \left( \frac\textbf + \frac\textbf + \frac\textbf \right) \cdot \textbf. Thus, we can set up the following flux integral as follows: : \begin I &=\iiint_V \nabla \cdot \mathbf \, \mathrmV\\ pt&=\iiint_V \left( \frac+\frac+\frac \right) \mathrmV\\ pt&=\iiint_V (4x+4y+4z) \, \mathrmV\\ pt&=\int_0^3 \int_^2 \int_0^ (4x+4y+4z) \, \mathrmV \end Now that we have set up the integral, we can evaluate it. :\begin \int_0^3 \int_^2 \int_0^ (4x+4y+4z) \, \mathrmV &=\int_^2 \int_0^ (12y+12z+18) \, \mathrmy \, \mathrmz\\ pt&=\int_0^ 24 (2z+3)\, \mathrmz\\ pt&=48\pi(2\pi+3) \end


Generalizations


Multiple dimensions

One can use the general Stokes' Theorem to equate the -dimensional volume integral of the divergence of a vector field over a region to the -dimensional surface integral of over the boundary of : : \underbrace_n \nabla \cdot \mathbf \, \mathrmV = \underbrace_ \mathbf \cdot \mathbf \, \mathrmS This equation is also known as the divergence theorem. When , this is equivalent to
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively orie ...
. When , it reduces to the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ...
, part 2.


Tensor fields

Writing the theorem in Einstein notation: : suggestively, replacing the vector field with a rank- tensor field , this can be generalized to: : 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 physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
see for example:
, and
).


See also

*
Kelvin–Stokes theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...


References


External links

*
Differential Operators and the Divergence Theorem
at MathPages
The Divergence (Gauss) Theorem
by Nick Bykov, Wolfram Demonstrations Project. * – ''This article was originally based on the
GFDL The GNU Free Documentation License (GNU FDL or simply GFDL) is a copyleft license for free documentation, designed by the Free Software Foundation (FSF) for the GNU Project. It is similar to the GNU General Public License, giving readers th ...
article from PlanetMath at https://web.archive.org/web/20021029094728/http://planetmath.org/encyclopedia/Divergence.html '' {{Calculus topics Theorems in calculus