
	Where results make sense

Topic: Divergence theorem

Related Topics

List of mathematical theorems

Divergence

Existence theorem

Fundamental theorem of calculus

Vector calculus

Mikhail Vasilievich Ostrogradsky

Integration by parts

Geohydrology

Derivative

Faraday cage

Finite element method

Examples of differential equations

Current (electricity)

List of ethnic enclaves in North American cities

Dynamic viscosity

	PlanetMath: divergence
		In physical terms, the divergence of a vector field is the extent to which the vector field flow behaves like a source or a sink at a given point.
		The non-infinitesimal interpretation of divergence is given by Gauss's Theorem.
		This theorem is a conservation law, stating that the volume total of all sinks and sources, i.e.
	planetmath.org /encyclopedia/DivergenceTheorem.html (251 words)

	Divergence Theorem
		Let V be a region in ⁫3complying with the hypotheses of the divergence theorem, and denote by S its boundary surface.
		By applying the divergence theorem to the vector field φc (1) show that: (∫∫∫v ▼φdV — ∫∫s φndS).c = 0 with the understanding that the integral of a vector is the vector of the integrals of the components.
		Green's, Divergence and Stokes theorems Describe in 5-15 lines the links and connections among Green's theorem in all forms, Stokes' theorem and the Divergence theorem.
	www.brainmass.com /homework-help/math/other/35020 (429 words)

	SparkNotes: Magnetic Field Theory: A Brief Review of Vector Calculus
		Thus the divergence is the sum of the partial differentials of the three functions that constitute the field.
		Speaking physically, the divergence of a vector field at a given point measures whether there is a net flow toward or away the point.
		A nonzero divergence indicates that at some point water is introduced or taken away from the system (a spring or a sinkhole).
	www.sparknotes.com /physics/magneticforcesandfields/magneticfieldtheory/section1.html (697 words)

	div.html
		Green's theorem expressed has a higher dimensional analog that is known as the divergence theorem.
		Points at which the divergence is negative are called sinks; points at which the divergence is positive are called sources.
		The divergence theorem, stated for solids bounded by a single closed surface, can be extended to solids bounded by several closed surfaces.
	steiner.math.nthu.edu.tw /disk3/cal03/html/div/div.html (1017 words)

	Amazon.co.uk: Books: Mathematical Methods for Physics and Engineering: A Comprehensive Guide (Site not responding. Last check: )
		DIVERGENCE THEOREM AND RELATED THEOREMS 11.8 Divergence theorem and related theorems The divergence theorem relates the total flux of a vector field out of a closed surface S to the integral...
		SURFACE AND VOLUME INTEGRALS 11.9 Stokes' theorem and related theorems Stokes' theorem is the `curl analogue' of the divergence theorem and relates the integral of the curl of a vector"
		(h) apply the divergence theorem to the volume bounded by the surface and the plane - - 0.
	www.amazon.co.uk /gp/reader/0521890675/ref=sib_books_ref/202-2023050-9304634?ie=UTF8&keywords=Divergence%20theorem&v=search-inside (276 words)

	Divergence theorem
		In vector calculus, the divergence theorem, also known as Gauss' theorem or Ostrogradsky-Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field.
		The divergence theorem is an important result for the mathematics of physics, in particular in electrostatics and fluid dynamics.
		The divergence theorem is thus a conservation law, stating that the volume total of all sinks and sources, i.e.
	www.askfactmaster.com /Divergence_theorem (639 words)

	divergence theorem
		In vector calculus, the divergence theorem, also known as Gauss' theorem or Ostrogradsky-Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field.
		The divergence theorem is an important result for the mathematics of physics, in particular in electrostatics and fluid dynamics.
		The theorem was first discovered by Joseph Louis Lagrange in 1762, then later independently rediscovered by Carl Friedrich Gauss in 1813, by George Green in 1825 and in 1831 by Mikhail Vasilievich Ostrogradsky, who also gave the first proof of the theorem.
	www.abacci.com /wikipedia/topic.aspx?cur_title=divergence_theorem (375 words)

	Amazon.co.uk: Books: Calculus: Multivariable
		The divergence of a magnetic vector field B must be zero everywhere.
		The gravitational field, F, of a planet of mass m at the origin is given by F =-GmIIrIls' Use the Divergence Theorem to show that the flux of the gravitational field through the sphere of radius a is independent...
		Then apply the Divergence Theorem to the region W lying between S and a small sphere SQ.] The divergence of a vector field is a sort of scalar derivative...
	www.amazon.co.uk /gp/reader/0471409529/ref=sib_books_ref/202-2023050-9304634?ie=UTF8&keywords=Divergence%20theorem&v=search-inside (287 words)

	Prof. Hans Wilhelm Alt: Script Analysis III -- Partial integration in Rn
		The proof of the "Divergence theorem" satz:6-7 will be reduced to the following two essential special cases.
		The first one is the local case in the interior of the domain Ω (see theorem satz:6-1).
		In the proof of the general divergence theorem we shall reduce the situation to the local ones in satz:6-1 and satz:6-3.
	www.iam.uni-bonn.de /~alt/ws2001/EN/analysis3-hyp_72.html (547 words)

	Results
		Times are for the divergence critic to speculate the lemmas and are for the average of 10 runs on a Sun 4 running QUINTUS 3.1.1.
		However, the divergence critic is able to identify the cause of this difficulty and propose a lemma which allows the proof to go through (example 15).
		To provide an indication of the difficulty of these theorems, the NQTHM system [8], which is perhaps the best known explicit induction theorem prover, was unable to prove more than half these theorems from the definitions alone.
	www.cs.cmu.edu /afs/cs/project/jair/pub/volume4/walsh96a-html/section3_8.html (1895 words)

	PlanetMath: Gauss Green theorem
		This theorem can be easily extended to piecewise regular domains.
		See Also: Green's theorem, general Stokes theorem, surface integration with respect to area, classical Stokes' theorem
		This is version 10 of Gauss Green theorem, born on 2005-02-11, modified 2005-02-18.
	planetmath.org /encyclopedia/GaussGreenTheorem.html (154 words)

	Divergence Theorem (Site not responding. Last check: )
		The divergence theorem in vector and tensor notation...
		EQUIVALENCE OF GREEN'S THEOREM AND THE DIVERGENCE THEOREM IN THE PLANE...
		Lab #4 Multiple Integrals and the Divergence Theorem MA 1024 P B00...
	www.scienceoxygen.com /math/514.html (227 words)

	Divergence Information
		In vector calculus, the divergence is an operator that measures a vector field's tendency to originate from or converge upon a given point.
		For instance, for a vector field that denotes the velocity of air expanding as it is heated, the divergence of the velocity field would have a positive value because the air is expanding.
		In physical terms, the divergence of a three dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point.
	www.bookrags.com /wiki/Divergence (628 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		In vector calculus, the divergence theorem, also known as Gauss' theorem, Ostrogradsky's theorem, or Gauss-Ostrogradsky theorem is a result that relates the outward flow of a vector field on a surface to the behaviour of the vector field inside the surface.
		More precisely, the divergence theorem states that the flux of a vector field on a surface is equal to the triple integral of the divergence on the region inside the surface.
		Note that the divergence theorem is a special case of the more general Stokes theorem which generalizes the fundamental theorem of calculus.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Divergence_theorem (810 words)

	Divergence theorem
		It was first discovered by Joseph Louis Lagrange (1736-1813) in 1762, then later independently rediscovered by Carl Friedrich Gauss (1777-1855) in 1813, by George Green (1793-1841) in 1825 and by Mikhail Vasilievich Ostrogradsky (1801-1862) in 1831, who also gave the first proof of the theorem.
		Let x,y,z be a system of Cartesian coordinates on a 3-dimensional Euclidean space, and let i,j,k be the corresponding basis of unit vectors.
		Indeed, an alternative, but logically equivalent definition, gives the divergence as the derivative of the net flow of the vector field across the surface of a small sphere relative to the surface area of the sphere.
	www.ebroadcast.com.au /lookup/encyclopedia/di/Divergence_theorem.html (366 words)

	Results
		Times are for the divergence critic to speculate the lemmas and are for the average of 10 runs on a Sun 4 running QUINTUS 3.1.1.
		However, the divergence critic is able to identify the cause of this difficulty and propose a lemma which allows the proof to go through (example 15).
		To provide an indication of the difficulty of these theorems, the NQTHM system [8], which is perhaps the best known explicit induction theorem prover, was unable to prove more than half these theorems from the definitions alone.
	www.cs.washington.edu /research/jair/volume4/walsh96a-html/section3_8.html (1895 words)

	Divergence Analysis
		The divergence critic instead studies the proof attempt looking for patterns of divergence; no attempt is made to analyse the rewrite rules themselves for structures which give rise to divergence.
		The divergence critic first partitions the sequence of equations which the prover attempts to prove by induction.
		The divergence analysis performed by the critic is summarised in Figure 1.
	www.cs.washington.edu /research/jair/volume4/walsh96a-html/section3_3.html (832 words)

	diver
		The divergence of a vector field at a point is defined as the net outward flux of that field per unit volume at that point.
		Divergence is useful in electromagnetics because, by utilizing the divergence theorem, we can often express a volume integral (a triple integral) by a surface integral (a double integral) and vice versa, sometimes simplifying our calculations.
		In addition, such use of the divergence theorem allows us to express one type of quantity in terms of another.
	members.tripod.com /llovesumi/diver.htm (668 words)

	Derivative operators
		Expressions for the derivative operators, such as gradient, divergence, curl, Laplacian, etc., are obtained by applying the Divergence Theorem to a differential volume increment bounded by coordinate surfaces.
		Although the equations 3.13 and 3.15 are equivalent expressions for the divergence, because of the identity 3.14, the numerical representations of these two forms may not be equivalent.
		It is important to note that since the conservative form of the divergence, and of the gradient and Laplacian to follow, is obtained directly from the closed surface integral in the Divergence Theorem, the use of the conservative difference forms for these derivative operators is equivalent to using difference forms for that closed surface integral.
	www.cse.ucsc.edu /~shreyas/btp/node25.html (495 words)

	Derivative operators
		Expressions for the derivative operators, such as gradient, divergence, curl, Laplacian, etc., are obtained by applying the Divergence Theorem to a differential volume increment bounded by coordinate surfaces.
		Although the equations 3.13 and 3.15 are equivalent expressions for the divergence, because of the identity 3.14, the numerical representations of these two forms may not be equivalent.
		It is important to note that since the conservative form of the divergence, and of the gradient and Laplacian to follow, is obtained directly from the closed surface integral in the Divergence Theorem, the use of the conservative difference forms for these derivative operators is equivalent to using difference forms for that closed surface integral.
	www.soe.ucsc.edu /~shreyas/btp/node25.html (495 words)

	[No title]
		This theorem can be used to describe the relationship between the way an incompressible fluid flows along or across the boundary of a plane and the way it moves inside the region.
		This is the first of two new ideas that we need for Green's theorem, and it is the flux density of a vector field at a point, called the divergence.
		This is the second of the two ideas that we need for Green's theorem, and it is the idea of circulation density of a vector field F at a point.
	faculty.eicc.edu /bwood/ma220supplemental/supplemental31.htm (859 words)

	Integrated Physics and Calculus Spring 2005
		Use the Divergence Theorem to rewrite the integral form of Gauss' Law as the differential form of Gauss' Law.
		The Divergence Theorem is almost free (particularly since we are being very casual about some of the limits involved).
		We'll see that Green's Theorem is a special case of Stokes' Theorem and that the Planar Divergence Theorem is an analog of the Divergence Theorem if we treat flux density as the ratio of flux through a curve to the area enclosed by the curve.
	www.math.ups.edu /~martinj/courses/spring2005/m221ph122/mathnotes.html (2609 words)

	Divergence Theorem
		This approach can be considered to arise from one of Maxwell's equations and involves the vector calculus operation called the divergence.
		The divergence of the electric field at a point in space is equal to the charge density divided by the permittivity of space.
		While these relationships could be used to calculate the electric field produced by a given charge distribution, the fact that E is a vector quantity increases the complexity of that calculation.
	hyperphysics.phy-astr.gsu.edu /hbase/electric/diverg.html (219 words)

	Vector Calculus
		The divergence of the curl is equal to zero:
		The volume integral of the divergence of a vector function is equal to the integral over the surface of the component normal to the surface.
		The area integral of the curl of a vector function is equal to the line integral of the field around the boundary of the area.
	hyperphysics.phy-astr.gsu.edu /hbase/vecal2.html (97 words)

	using the divergence theorem (Site not responding. Last check: )
		However, it sometimes is, and this is a nice example of both the divergence theorem and a flux integral, so we'll go through it as is.
		Because this is not a closed surface, we can't use the divergence theorem to evaluate the flux integral.
		Using the divergence theorem, we get the value of the flux through the top and bottom surface together to be 5 pi / 3, and the flux calculation for the bottom surface gives zero, so that the flux just through the top surface is also 5 pi / 3.
	www-personal.umich.edu /~glarose/classes/calcIII/web/17_9 (418 words)

	[No title]
		Mathematicians were not immune, and at a mathematics conference in July, 1999, Paul and Jack Abad presented their list of "The Hundred Greatest Theorems." Their ranking is based on the following criteria: "the place the theorem holds in the literature, the quality of the proof, and the unexpectedness of the result."
		The list is of course as arbitrary as the movie and book list, but the theorems here are all certainly worthy results.
		Liouville’s Theorem and the Construction of Trancendental Numbers
	personal.stevens.edu /~nkahl/Top100Theorems.html (217 words)

	[No title]
		Slide 4 — Reynolds Transport Theorem Lets now start by letting V (with a bar through it), which is a function of time, be an arbitrary fluid material volume (arbitrary is important here).
		It does not matter which it is. The proof of the theorem does not change, and the applicability of the theorem is the same to either scalar or vector functions.
		That can be rewritten using the divergence theorem as the integral over the surface, which bounds the volume, capital V of lowercase t, of the quantity, capital F times lowercase v dot n lowercase d capital S, a normal component of that quantity.
	www.catea.org /grade/mecheng/Text/Module6.doc (1988 words)

Try your search on: Qwika (all wikis)