Divergence theorem

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In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem relating the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed.

More precisely, the divergence theorem states that the surface integral 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 enclosed by 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 and engineering, particularly in electrostatics and fluid dynamics. 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 the fundamental theorem of calculus. In two dimensions, it is equivalent to Green's theorem.

Explanation using liquid flow

Template:See also Vector fields are often illustrated using the example of the velocity 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 at any time is equal to the volume rate of fluid crossing this surface, i.e., the surface integral 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 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

File:Divergence theorem.svg

A region Template:Mvar bounded by the surface $S=\partial V$ with the surface normal Template:Mvar

Suppose Template:Mvar is a subset of ${ℝ}^n$ (in the case of Template:Math represents a volume in three-dimensional space) which is compact and has a piecewise smooth boundary Template:Mvar (also indicated with $\partial V=S$). If Template:Math is a continuously differentiable vector field defined on a neighborhood of Template:Mvar, then:

Template:Oiint

The left side is a volume integral over the volume Template:Mvar, and the right side is the surface integral over the boundary of the volume Template:Mvar. The closed, measurable set $\partial V$ is oriented by outward-pointing normals, and $\widehat{𝐧}$ is the outward pointing unit normal at almost each point on the boundary $\partial V$. ($\mathrm{d}𝐒$ may be used as a shorthand for $𝐧\mathrm{d}S$.) 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 Template:Mvar. Template:Clear

Informal derivation

The divergence theorem follows from the fact that if a volume Template:Mvar is partitioned into separate parts, the flux out of the original volume is equal to the algebraic 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.

A volume divided into two subvolumes. At right the two subvolumes are separated to show the flux out of the different surfaces.

See the diagram. A closed, bounded volume Template:Mvar is divided into two volumes Template:Math and Template:Math by a surface Template:Math (green). The flux Template:Math out of each component region Template:Math is equal to the sum of the flux through its two faces, so the sum of the flux out of the two parts is

$\mathrm{\Phi }(V_{\text{1}})+\mathrm{\Phi }(V_{\text{2}})={\mathrm{\Phi }}_{\text{1}}+{\mathrm{\Phi }}_{\text{31}}+{\mathrm{\Phi }}_{\text{2}}+{\mathrm{\Phi }}_{\text{32}}$

where Template:Math and Template:Math are the flux out of surfaces Template:Math and Template:Math, Template:Math is the flux through Template:Math out of volume 1, and Template:Math is the flux through Template:Math out of volume 2. The point is that surface Template:Math is part of the surface of both volumes. The "outward" direction of the normal vector $\widehat{𝐧}$ is opposite for each volume, so the flux out of one through Template:Math is equal to the negative of the flux out of the other so these two fluxes cancel in the sum.

${\mathrm{\Phi }}_{\text{31}}=\underset{S_3}{\iint }𝐅\cdot \widehat{𝐧}\mathrm{d}S=-\underset{S_3}{\iint }𝐅\cdot (-\widehat{𝐧})\mathrm{d}S=-{\mathrm{\Phi }}_{\text{32}}$

Therefore:

$\mathrm{\Phi }(V_{\text{1}})+\mathrm{\Phi }(V_{\text{2}})={\mathrm{\Phi }}_{\text{1}}+{\mathrm{\Phi }}_{\text{2}}$

Since the union of surfaces Template:Math and Template:Math is Template:Mvar

$\mathrm{\Phi }(V_{\text{1}})+\mathrm{\Phi }(V_{\text{2}})=\mathrm{\Phi }(V)$

Error creating thumbnail:

The volume can be divided into any number of subvolumes and the flux out of V is equal to the sum of the flux out of each subvolume, because the flux through the green surfaces cancels out in the sum. In (b) the volumes are shown separated slightly, illustrating that each green partition is part of the boundary of two adjacent volumes

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.

$\mathrm{\Phi }(V)=\sum _{V_{\text{i}}\subset V}\mathrm{\Phi }(V_{\text{i}})$

File:Divergence theorem 3 - infinitesimals.png

As the volume is subdivided into smaller parts, the ratio of the flux $\mathrm{\Phi }(V_{\text{i}})$ out of each volume to the volume $|V_{\text{i}}|$ approaches $\mathrm{div}𝐅$

The flux Template:Math out of each volume is the surface integral of the vector field Template:Math over the surface

$\underset{S(V)}{\iint }𝐅\cdot \widehat{𝐧}\mathrm{d}S=\sum _{V_{\text{i}}\subset V}\underset{S(V_{\text{i}})}{\iint }𝐅\cdot \widehat{𝐧}\mathrm{d}S$

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 Template:Math approaches zero. However, from the definition of divergence, the ratio of flux to volume, $\frac{\mathrm{\Phi }(V_{\text{i}})}{|V_{\text{i}}|}=\frac{1}{|V_{\text{i}}|}\underset{S(V_{\text{i}})}{\iint }𝐅\cdot \widehat{𝐧}\mathrm{d}S$, the part in parentheses below, does not in general vanish but approaches the divergence Template:Math as the volume approaches zero.

$\underset{S(V)}{\iint }𝐅\cdot \widehat{𝐧}\mathrm{d}S=\sum _{V_{\text{i}}\subset V}(\frac{1}{|V_{\text{i}}|}\underset{S(V_{\text{i}})}{\iint }𝐅\cdot \widehat{𝐧}\mathrm{d}S)|V_{\text{i}}|$

As long as the vector field Template:Math has continuous derivatives, the sum above holds even in the limit when the volume is divided into infinitely small increments

$\underset{S(V)}{\iint }𝐅\cdot \widehat{𝐧}\mathrm{d}S=\underset{|V_{\text{i}}|\to 0}{\lim }\sum _{V_{\text{i}}\subset V}(\frac{1}{|V_{\text{i}}|}\underset{S(V_{\text{i}})}{\iint }𝐅\cdot \widehat{𝐧}\mathrm{d}S)|V_{\text{i}}|$

As $|V_{\text{i}}|$ approaches zero volume, it becomes the infinitesimal Template:Math, the part in parentheses becomes the divergence, and the sum becomes a volume integral over Template:Mvar

$\underset{S(V)}{\iint }𝐅\cdot \widehat{𝐧}\mathrm{d}S=\underset{V}{\iiint }\mathrm{div}𝐅\mathrm{d}V$

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:Template:Citation needed Template:Math theorem

Proof of Theorem. Template:Ordered list

For compact Riemannian manifolds with boundary

We are going to prove the following:Template:Citation needed Template:Math theorem

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 $O\subset \bar{\mathrm{\Omega }}$. First consider the case where the patch is disjoint from $\partial \mathrm{\Omega }$. Then $O$ is identified with an open subset of ${ℝ}^n$ and integration by parts produces no boundary terms: $$\begin{array}{cc}(\mathrm{grad}u,X)& =\underset{O}{\int }⟨\mathrm{grad}u,X⟩\sqrt{g}dx\\ & =\underset{O}{\int }{\partial }_{j}u{X}^j\sqrt{g}dx\\ & =-\underset{O}{\int }u{\partial }_j(\sqrt{g}{X}^j)dx\\ & =-\underset{O}{\int }u\frac{1}{\sqrt{g}}{\partial }_j(\sqrt{g}{X}^j)\sqrt{g}dx\\ & =(u,-\frac{1}{\sqrt{g}}{\partial }_j(\sqrt{g}{X}^j))\\ & =(u,-\mathrm{div}X).\end{array}$$ In the last equality we used the Voss-Weyl coordinate formula for the divergence, although the preceding identity could be used to define $-\mathrm{div}$ as the formal adjoint of $\mathrm{grad}$. Now suppose $O$ intersects $\partial \mathrm{\Omega }$. Then $O$ is identified with an open set in ${ℝ}_{+}^n=\{x\in {ℝ}^n:x_n\ge 0\}$. We zero extend $u$ and $X$ to ${ℝ}_{+}^n$ and perform integration by parts to obtain $$\begin{array}{cc}(\mathrm{grad}u,X)& =\underset{O}{\int }⟨\mathrm{grad}u,X⟩\sqrt{g}dx\\ & =\underset{{ℝ}_{+}^n}{\int }{\partial }_{j}u{X}^j\sqrt{g}dx\\ & =(u,-\mathrm{div}X)-\underset{{ℝ}^{n-1}}{\int }u(x^{\prime },0)X^n(x^{\prime },0)\sqrt{g(x^{\prime },0)}d{x}^{\prime },\end{array}$$ where $d{x}^{\prime }=d{x}_1\dots d{x}_{n-1}$. By a variant of the straightening theorem for vector fields, we may choose $O$ so that $\frac{\partial }{\partial x_n}$ is the inward unit normal $-N$ at $\partial \mathrm{\Omega }$. In this case $\sqrt{g(x^{\prime },0)}d{x}^{\prime }=\sqrt{g_{\partial \mathrm{\Omega }}(x^{\prime })}d{x}^{\prime }=dS$ is the volume element on $\partial \mathrm{\Omega }$ and the above formula reads $$(\mathrm{grad}u,X)=(u,-\mathrm{div}X)+\underset{\partial \mathrm{\Omega }}{\int }u⟨X,N⟩dS.$$ This completes the proof.

Corollaries

By replacing Template:Math in the divergence theorem with specific forms, other useful identities can be derived (cf. vector identities).

• With $𝐅\to 𝐅g$ for a scalar function Template:Mvar and a vector field Template:Math,

Template:Oiint

A special case of this is $𝐅=\mathrm{\nabla }f$, in which case the theorem is the basis for Green's identities.

• With $𝐅\to 𝐅\times 𝐆$ for two vector fields Template:Math and Template:Math, where $\times$ denotes a cross product,

Template:Oiint

• With $𝐅\to 𝐅\cdot 𝐆$ for two vector fields Template:Math and Template:Math, where $\cdot$ denotes a dot product,

Template:Oiint

• With $𝐅\to f𝐜$ for a scalar function Template:Math and vector field c:

Template:Oiint

The last term on the right vanishes for constant $𝐜$ 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 $𝐜$ to be constant:

Template:Oiint

• With $𝐅\to 𝐜\times 𝐅$ for vector field Template:Math and constant vector c:

Template:Oiint

By reordering the triple product on the right hand side and taking out the constant vector of the integral,

Template:Oiint

Hence,

Template:Oiint

Example

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

File:SurfacesWithAndWithoutBoundary.svg

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 we wish to evaluate

Template:Oiint

where Template:Mvar is the unit sphere defined by

$S=\{(x,y,z)\in {ℝ}^3:x^2+y^2+z^2=1\},$

and Template:Math is the vector field

$𝐅=2x𝐢+y^2𝐣+z^2𝐤.$

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:

$\underset{W}{\iiint }(\mathrm{\nabla }\cdot 𝐅)\mathrm{d}V=2\underset{W}{\iiint }(1+y+z)\mathrm{d}V=2\underset{W}{\iiint }\mathrm{d}V+2\underset{W}{\iiint }y\mathrm{d}V+2\underset{W}{\iiint }z\mathrm{d}V,$

where Template:Mvar is the unit ball:

$W=\{(x,y,z)\in {ℝ}^3:x^2+y^2+z^2\le 1\}.$

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:

$\underset{W}{\iiint }y\mathrm{d}V=\underset{W}{\iiint }z\mathrm{d}V=0.$

Therefore,

Template:Oiint

because the unit ball Template:Mvar has volume Template:Math.

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 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).

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.

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, 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 $R$:

$R=\{(x,y)\in {ℝ}^2:x^2+y^2\le 1\},$

and the vector field:

$𝐅(x,y)=2y𝐢+5x𝐣.$

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

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

such that $0\le s\le 2\pi$ 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

$C(s)=\cos (s)𝐢+\sin (s)𝐣.$

At a point $P$ on $C$:

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

Therefore,

$\begin{array}{cc}{\oint }_C𝐅\cdot 𝐧\mathrm{d}s& =\underset{0}{\overset{2\pi }{\int }}(2\sin (s)𝐢+5\cos (s)𝐣)\cdot (\cos (s)𝐢+\sin (s)𝐣)\mathrm{d}s\\ & =\underset{0}{\overset{2\pi }{\int }}(2\sin (s)\cos (s)+5\sin (s)\cos (s))\mathrm{d}s\\ & =7\underset{0}{\overset{2\pi }{\int }}\sin (s)\cos (s)\mathrm{d}s\\ & =0.\end{array}$

Because $M=\Re 𝔢(𝐅)=2y$, we can evaluate Template:Nowrap and because Template:Nowrap $\frac{\partial N}{\partial y}=0$. Thus

$\underset{R}{\iint }\mathrm{\nabla }\cdot 𝐅\mathrm{d}A=\underset{R}{\iint }(\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y})\mathrm{d}A=0.$

Example 2

Let's say we wanted to evaluate the flux of the following vector field defined by $𝐅=2x^2\mathbf{\text{i}}+2y^2\mathbf{\text{j}}+2z^2\mathbf{\text{k}}$ bounded by the following inequalities:

$\{0\le x\le 3\},\{-2\le y\le 2\},\{0\le z\le 2\pi \}$

By the divergence theorem,

Template:Oiint

We now need to determine the divergence of $\mathbf{\text{F}}$. If $𝐅$ is a three-dimensional vector field, then the divergence of $\mathbf{\text{F}}$ is given by $\mathrm{\nabla }\cdot \mathbf{\text{F}}=(\frac{\partial }{\partial x}\mathbf{\text{i}}+\frac{\partial }{\partial y}\mathbf{\text{j}}+\frac{\partial }{\partial z}\mathbf{\text{k}})\cdot \mathbf{\text{F}}$.

Thus, we can set up the following flux integral Template:Oiint as follows:

$\begin{array}{cc}I& =\underset{V}{\iiint }\mathrm{\nabla }\cdot 𝐅\mathrm{d}V\\ & =\underset{V}{\iiint }(\frac{\partial {𝐅}_{𝐱}}{\partial x}+\frac{\partial {𝐅}_{𝐲}}{\partial y}+\frac{\partial {𝐅}_{𝐳}}{\partial z})\mathrm{d}V\\ & =\underset{V}{\iiint }(4x+4y+4z)\mathrm{d}V\\ & =\underset{0}{\overset{3}{\int }}\underset{-2}{\overset{2}{\int }}\underset{0}{\overset{2\pi }{\int }}(4x+4y+4z)\mathrm{d}V\end{array}$

Now that we have set up the integral, we can evaluate it.

$\begin{array}{cc}\underset{0}{\overset{3}{\int }}\underset{-2}{\overset{2}{\int }}\underset{0}{\overset{2\pi }{\int }}(4x+4y+4z)\mathrm{d}V& =\underset{-2}{\overset{2}{\int }}\underset{0}{\overset{2\pi }{\int }}(12y+12z+18)\mathrm{d}y\mathrm{d}z\\ & =\underset{0}{\overset{2\pi }{\int }}24(2z+3)\mathrm{d}z\\ & =48\pi (2\pi +3)\end{array}$

Generalizations

Multiple dimensions

One can use the generalised Stokes' theorem to equate the Template:Mvar-dimensional volume integral of the divergence of a vector field Template:Math over a region Template:Mvar to the Template:Math-dimensional surface integral of Template:Math over the boundary of Template:Mvar:

$\underset{n}{\underbrace{\int \cdots \underset{U}{\int }}}\mathrm{\nabla }\cdot 𝐅\mathrm{d}V=\underset{n-1}{\underbrace{{\oint }_{}\cdots {\oint }_{\partial U}}}𝐅\cdot 𝐧\mathrm{d}S$

This equation is also known as the divergence theorem.

When Template:Math, this is equivalent to Green's theorem.

When Template:Math, it reduces to the fundamental theorem of calculus, part 2.

Tensor fields

Writing the theorem in Einstein notation:

Template:Oiint

suggestively, replacing the vector field Template:Math with a rank-Template:Mvar tensor field Template:Mvar, this can be generalized to:

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).

See also

• Kelvin–Stokes theorem
• Generalized Stokes theorem
• Differential form

External links

• Template:Springer
• Differential Operators and the Divergence Theorem at MathPages
• The Divergence (Gauss) Theorem by Nick Bykov, Wolfram Demonstrations Project.
• Template:MathWorld – This article was originally based on the GFDL article from PlanetMath at https://web.archive.org/web/20021029094728/http://planetmath.org/encyclopedia/Divergence.html