|
|
| Line 1: |
Line 1: |
| {{redirect|Gauss's theorem|Gauss's theorem concerning the electric field|Gauss's law}}
| | I'm Fallon (21) from Snesudden, Sweden. <br>I'm learning Russian literature at a local high school and I'm just about to graduate.<br>I have a part time job in a university.<br><br>Look at my page health ([http://www.getflatulencehelp.com/ www.getflatulencehelp.com]) |
| {{Calculus |Vector}}
| |
| | |
| In [[vector calculus]], the '''divergence theorem''', also known as '''Gauss's theorem''' or '''Ostrogradsky's theorem''',<ref>or less correctly as '''[[Carl Friedrich Gauss|Gauss]]' theorem'''<!--apostrophe is correct.--> (see history for reason)</ref><ref name="Katz">{{cite journal
| |
| | last = Katz
| |
| | first = Victor J.
| |
| | title = The history of Stokes's theorem
| |
| | journal = Mathematics Magazine
| |
| | volume = 52
| |
| | issue =
| |
| | pages = 146–156
| |
| | publisher = Mathematical Association of America
| |
| | year = 1979
| |
| | url =
| |
| | issn =
| |
| | doi =
| |
| | id =
| |
| }} reprinted in {{cite book
| |
| | last = Anderson
| |
| | first = Marlow
| |
| | title = Who Gave You the Epsilon?: And Other Tales of Mathematical History
| |
| | publisher = Mathematical Association of America
| |
| | year = 2009
| |
| | location =
| |
| | pages = 78–79
| |
| | url = http://books.google.com/books?id=WwFMjsym9JwC&pg=PA78&dq=%22ostrogradsky's+theorem
| |
| | doi =
| |
| | id =
| |
| | isbn = 0883855690}}</ref> 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]].<ref>{{Citation
| |
| | last = Stewart | first = James | author-link = James Stewart (mathematician) | title = Calculus: Early Transcendentals | publisher = Thomson Brooks/Cole | year = 2008 | edition = 6 | chapter = Vector Calculus | isbn = 978-0-495-01166-8}}</ref>
| |
| | |
| == 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.<ref>{{cite book | author = R. G. Lerner, G. L. Trigg |edition = 2nd | title = Encyclopaedia of Physics | publisher = VHC | year = 1994 | isbn = 3-527-26954-1 }}</ref>
| |
| | |
| 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.<ref>{{Citation | last1 = Byron | first1 = Frederick | last2 = Fuller | first2 = Robert | author2-link = Robert W. Fuller | title = Mathematics of Classical and Quantum Physics | publisher = Dover Publications | year = 1992 | page = 22 | isbn = 978-0-486-67164-2 }}</ref>
| |
| | |
| ==Mathematical statement==
| |
| [[File:Divergence theorem.svg|thumb|right|250px|A region ''V'' bounded by the surface ''S''=∂''V'' with the surface normal ''n'']]
| |
| [[File:SurfacesWithAndWithoutBoundary.svg|right|thumb|200px|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 ''V'' is a subset of '''R'''<sup>''n''</sup> (in the case of ''n'' = 3, ''V'' represents a volume in 3D space) which is [[compact space|compact]] and has a [[piecewise]] [[smooth surface|smooth boundary]] S (also indicated with ∂''V''=S). If '''F''' is a continuously differentiable vector field defined on a neighborhood of ''V'', then we have:<ref name=spiegel>{{cite book | author = M. R. Spiegel; S. Lipschutz; D. Spellman | title = Vector Analysis | edition = 2nd | series = Schaum’s Outlines | publisher = McGraw Hill | location = USA | year = 2009 | isbn = 978-0-07-161545-7 }}</ref>
| |
| | |
| :{{oiint
| |
| | preintegral = <math>\int\!\!\!\!\int\!\!\!\!\int_V\left(\mathbf{\nabla}\cdot\mathbf{F}\right)dV=</math>
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>(\mathbf{F}\cdot\mathbf{n})\,dS .</math>
| |
| }}
| |
| | |
| 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 [[Surface normal|normals]], and '''n''' is the outward pointing unit normal field of the boundary ∂''V''. (d'''S''' may be used as a shorthand for '''n''' d''S''.) 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]]).<ref name=spiegel/>
| |
| | |
| * Applying the divergence theorem to the product of a scalar function ''g'' and a vector field '''F''', the result is
| |
| | |
| :{{oiint
| |
| | preintegral = <math>\int\!\!\!\!\int\!\!\!\!\int_V\left[\mathbf{F}\cdot \left(\nabla g\right) + g \left(\nabla\cdot \mathbf{F}\right)\right] dV
| |
| =</math>
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>g\mathbf{F}\cdot \mathbf{n} \,dS.</math>
| |
| }}
| |
| | |
| :A special case of this is <math>\scriptstyle \mathbf{F}=\nabla f</math>, in which case the theorem is the basis for [[Green's identities]].
| |
| | |
| * Applying the divergence theorem to the cross-product of two vector fields <math>\scriptstyle \mathbf{F}\times \mathbf{G}</math>, the result is
| |
| | |
| ::{{oiint
| |
| | preintegral = <math>\int\!\!\!\!\int\!\!\!\!\int_V\left[\mathbf{G}\cdot\left(\nabla\times\mathbf{F}\right) - \mathbf{F}\cdot \left( \nabla\times\mathbf{G}\right)\right]\, dV =</math>
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>\mathbf F\times\mathbf{G}\cdot d\mathbf{S}.</math>
| |
| }}
| |
| | |
| * Applying the divergence theorem to the product of a scalar function, ''f'', and a non-zero constant vector, the following theorem can be proven:<ref name=mathworld>[http://mathworld.wolfram.com/DivergenceTheorem.html MathWorld]</ref>
| |
| | |
| ::{{oiint
| |
| | preintegral = <math>\int\!\!\!\!\int\!\!\!\!\int_V\nabla f\, dV =</math>
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>f d\mathbf{S}.</math>
| |
| }}
| |
| | |
| * 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:<ref name=mathworld/>
| |
| | |
| ::{{oiint
| |
| | preintegral = <math>\int\!\!\!\!\int\!\!\!\!\int_V\nabla\times\mathbf{F}\, dV =</math>
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>d \mathbf{S}\times \mathbf{F}.</math>
| |
| }}
| |
| | |
| == Example ==
| |
| | |
| [[File:Vector Field on a Sphere.png|thumb|The vector field corresponding to the example shown. Note, vectors may point into or out of the sphere.]]
| |
| | |
| Suppose we wish to evaluate
| |
| | |
| :{{oiint
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>\mathbf{F}\cdot\mathbf{n} \, dS,</math>
| |
| }}
| |
| | |
| where ''S'' is the [[unit sphere]] defined by
| |
| | |
| :<math> x^2+y^2+z^2=1</math>
| |
| | |
| and '''F''' is the [[vector field]]
| |
| | |
| :<math>\mathbf{F} = 2 x\mathbf{i}+y^2\mathbf{j}+z^2\mathbf{k}.</math>
| |
| | |
| The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem:
| |
| | |
| :{|
| |
| |- valign="top"
| |
| | {{oiint
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>\mathbf{F}\cdot\mathbf{n} \, dS</math>
| |
| }} || || <math>
| |
| \begin{align}
| |
| &= \int\!\!\!\!\int\!\!\!\!\int_W\left(\nabla\cdot\mathbf{F}\right) \, dV\\
| |
| &= 2\int\!\!\!\!\int\!\!\!\!\int_W\left(1+y+z\right) \, dV\\
| |
| &= 2\int\!\!\!\!\int\!\!\!\!\int_W \,dV + 2\int\!\!\!\!\int\!\!\!\!\int_W y \,dV + 2\int\!\!\!\!\int\!\!\!\!\int_W z \,dV.
| |
| \end{align}
| |
| </math>
| |
| |}
| |
| | |
| where ''W'' is the unit ball (i.e., the interior and surface of the unit sphere, <math>\scriptstyle x^2+y^2+z^2\leq 1</math>). Since the function <math>\scriptstyle y</math> is positive in one hemisphere of ''W'' and negative in the other, in an equal and opposite way, its total integral over ''W'' is zero. The same is true for <math>\scriptstyle z</math>:
| |
| | |
| :<math>\int\!\!\!\int\!\!\!\int_W y\, dV = \int\!\!\!\int\!\!\!\int_W z\, dV = 0.</math>
| |
| | |
| Therefore,
| |
| | |
| :{{oiint
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>\mathbf{F}\cdot\mathbf{n}\,{d}S = 2\int\!\!\!\!\int\!\!\!\!\int_W\, dV = \frac{8\pi}{3},</math>
| |
| }}
| |
| | |
| because the unit ball W has [[volume]] <math>\scriptstyle \frac{4\pi}{3}.</math>
| |
| | |
| ==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====
| |
| {{main|continuity equation}}
| |
| | |
| [[Continuity equation]]s 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 equation]]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).<ref name="C.B. Parker 1994">{{cite book| author=C.B. Parker|edition=2nd|title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill| year=1994 | isbn=0-07-051400-3}}</ref>
| |
| | |
| ===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.<ref name="C.B. Parker 1994"/>
| |
| | |
| == History ==
| |
| The [[theorem]] was first discovered by [[Joseph Louis Lagrange|Lagrange]] in 1762,<ref>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: [http://books.google.com/books?id=3TA4DeQw1NoC&pg=PA151#v=onepage&q&f=false "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; [http://books.google.com/books?id=3TA4DeQw1NoC&pg=PA263#v=onepage&q&f=false on pages 263-265], Lagrange transforms triple integrals into double integrals using integration by parts.</ref> then later independently rediscovered by [[Carl Friedrich Gauss|Gauss]] in 1813,<ref>C. F. Gauss (1813) [http://books.google.com.sv/books?id=ASwoAQAAMAAJ&pg=PP355#v=onepage&q&f=false "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.</ref> by [[Mikhail Vasilievich Ostrogradsky|Ostrogradsky]], who also gave the first proof of the general theorem, in 1826,<ref>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) [http://books.google.com/books?id=XXMhAQAAMAAJ&pg=PA129#v=onepage&q&f=false "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) [http://www-personal.umich.edu/~madeland/math255/files/Stokes-Katz.pdf "The history of Stokes' theorem,"] ''Mathematics Magazine'', '''52'''(3): 146-156; for Ostragradsky's proof of the divergence theorem, see pages 147-148.</ref> by [[George Green|Green]] in 1828,<ref>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 [http://books.google.com/books?id=GwYXAAAAYAAJ&pg=PA10#v=onepage&q&f=false pages 10-12].</ref> etc.<ref>Other early investigators who used some form of the divergence theorem include:
| |
| *[[Siméon Denis Poisson|Poisson]] (presented: February 2, 1824 ; published: 1826) [http://gallica.bnf.fr/ark:/12148/bpt6k3220m/f255.image "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.
| |
| * [[Pierre Frédéric Sarrus|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.</ref> 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 ''R'', where
| |
| | |
| :<math> \mathbf{F}(x,y)=2 y\mathbf{i} + 5x \mathbf{j} ,\ </math>
| |
| | |
| and ''R'' is the region bounded by the circle
| |
| | |
| :<math> x^2 + y^2 = 1.\ </math>
| |
| | |
| The boundary of ''R'' is the unit circle, ''C'', that can be represented parametrically by:
| |
| | |
| :<math>x\, =\cos(s);\ y\, = \sin(s)\ </math>
| |
| | |
| such that <math>\scriptstyle 0\, \le \,s\, \le \,2\pi</math> 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
| |
| | |
| :<math>\mathbf{C}(s) = \cos(s)\mathbf{i} + \sin(s)\mathbf{j}.\ </math>
| |
| | |
| At a point '''P''' on ''C'':
| |
| | |
| :<math> \mathbf{P}\, = \,(\cos(s),\, \sin(s)) \, \Rightarrow \, \mathbf{F}\, = \,2\sin(s)\mathbf{i} + 5\cos(s)\mathbf{j} \, .</math>
| |
| | |
| Therefore,
| |
| | |
| :<math>\begin{align}\oint_C \mathbf{F} \cdot \mathbf{n}\, ds &= \,\int_{0}^{2 \pi} ( 2 \sin s \mathbf{i} + 5 \cos s \mathbf{j}) \cdot (\cos s \mathbf{i} + \sin s \mathbf{j})\, ds\\
| |
| &= \,\int_{0}^{2 \pi} (2 \sin s \cos s + 5 \sin s \cos s)\, ds\\
| |
| &= \,7\int_{0}^{2 \pi} \sin s \cos s\, ds\\
| |
| &= \,0.\end{align}\ </math>
| |
| | |
| Because <math>\scriptstyle M\, = \,2y\,</math>, <math>\scriptstyle \frac{\partial M}{\partial x} = \,0</math>, and because <math>\scriptstyle N\, = \,5x\,</math>, <math>\scriptstyle \frac{\partial N}{\partial y} = \,0</math>. Thus
| |
| | |
| :<math>\iint\limits_R \, \operatorname{div} \mathbf{F} \,dA = \iint\limits_R \left (\frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} \right) \, dA = 0. </math>
| |
| | |
| ==Generalizations==
| |
| ===Multiple dimensions===
| |
| One can use the general [[Stokes' Theorem]] to equate the n-dimensional volume integral of the divergence of a vector field F over a region U to the (n-1)-dimensional surface integral of F over the boundary of U:
| |
| | |
| <math> \int_U \nabla \cdot \mathbf{F} \, dV_n = \oint_{\partial U} \mathbf{F} \cdot \mathbf{n} \, dS_{n-1} </math>
| |
| | |
| 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===
| |
| | |
| {{main|Tensor field}}
| |
| | |
| Writing the theorem in [[Einstein notation]]:
| |
| | |
| :{{oiint
| |
| | preintegral = <math>\int\!\!\!\!\int\!\!\!\!\int_V \dfrac{\partial F_i}{\partial x_i}dV=</math>
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>F_i n_i\,dS </math>
| |
| }}
| |
| | |
| suggestively, replacing the vector field ''F'' with a rank-''n'' tensor field ''T'', this can be generalized to:<ref>{{cite book| author=K.F. Riley, M.P. Hobson, S.J. Bence| title=Mathematical methods for physics and engineering| publisher=Cambridge University Press| year=2010 | isbn=978-0-521-86153-3}}</ref>
| |
| | |
| :{{oiint
| |
| | preintegral = <math>\int\!\!\!\!\int\!\!\!\!\int_V \dfrac{\partial T_{i_1i_2\cdots i_q\cdots i_n}}{\partial x_{i_q}}dV=</math>
| |
| | intsubscpt = <math>\scriptstyle S</math>
| |
| | integrand = <math>T_{i_1i_2\cdots i_q\cdots i_n}n_{i_q}\,dS .</math>
| |
| }}
| |
| | |
| 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]]<ref>see for example: <br />{{cite book |pages=85–86, §3.5| author=J.A. Wheeler, C. Misner, K.S. Thorne| title=[[Gravitation (book)|Gravitation]]| publisher=W.H. Freeman & Co| year=1973 | isbn=0-7167-0344-0}}, and <br />{{cite book |author=R. Penrose| title=[[The Road to Reality]]| publisher= Vintage books| year=2007 | isbn=0-679-77631-1}}</ref>).
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==External links==
| |
| * [http://www.encyclopediaofmath.org/index.php/Divergence_theorem Divergence theorem] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
| |
| * [http://www.mathpages.com/home/kmath330/kmath330.htm Differential Operators and the Divergence Theorem] at MathPages
| |
| * [http://demonstrations.wolfram.com/TheDivergenceGaussTheorem/ The Divergence (Gauss) Theorem] by Nick Bykov, [[Wolfram Demonstrations Project]].
| |
| * {{MathWorld |title=Divergence Theorem |urlname=DivergenceTheorem}}
| |
| —
| |
| ''This article was originally based on the [[GFDL]] article from [[PlanetMath]] at http://planetmath.org/encyclopedia/Divergence.html ''
| |
| | |
| [[Category:Theorems in calculus]]
| |