Find Physics Book Answers
Help w/Divergence theorem Step by Step Physics Solutions.
www.Cramster.com/physicshwanswers

Divergence theorem - Wikipedia, the free encyclopedia
More precisely, the divergence theorem states that the outward flux of a vector ... The divergence theorem is an important result for the mathematics of engineering, ...
en.wikipedia.org

Divergence Theorem -- from Wolfram MathWorld
The divergence theorem is a mathematical statement of the physical fact that, in ... Divergence Theorem." From MathWorld--A Wolfram Web Resource. ...
mathworld.wolfram.com

Divergence Theorem
Part of the Proof of the Divergence Theorem ... We also note that the divergence theorem can be written as. We will show that ...
ltcconline.net

Pauls Online Notes : Calculus III - Divergence Theorem
Online Notes / Calculus III / Surface Integrals / Divergence Theorem. Calculus III ... Example 1 Use the divergence theorem to evaluate where and the surface consists ...
tutorial.math.lamar.edu

Differential Operators and the Divergence Theorem
Hence we can express the Divergence Theorem in its familiar form ... Another important consequence of the Divergence Theorem can be seen by noting ...
www.mathpages.com

divergence definition | Dictionary.com
Definition of divergence at Dictionary.com with free audio pronunciation. ... divergence of a vector field. what is genetic divergence. proof of divergence theorem ...
dictionary.reference.com

Divergence Theorem
Divergence Theorem. This posting is for problem #5.15. See attached file for full problem description. ... Divergence Theorem, Unit conversion - Divergence ...
www.brainmass.com

Divergence - Wikipedia, the free encyclopedia
... flow outwards of a region is made precise by the divergence theorem. ... Divergence theorem. Curl. Gradient. Nabla in cylindrical and spherical coordinates ...
en.wikipedia.org

MATH 2423 #15: DIVERGENCE THEOREM AND STOKE'S THEOREM
... called the Divergence Theorem. To use the Divergence Theorem, the vector ... x 2 + y 2 + z 2 = 4. Is this the same answer you got using the Divergence Theorem? ...
www-math.cudenver.edu

Warning: mkdir() [function.mkdir]: Permission denied in /home/webs/affiliatelib2/CacheManager.php on line 12

Warning: mkdir() [function.mkdir]: No such file or directory in /home/webs/affiliatelib2/CacheManager.php on line 12

Warning: fopen(/home/templatecore2cache//*cluesnet.com/4c/4c81c645bc33a20a3e65af6d3afcd2dcb5530ec7.tc2cache) [function.fopen]: failed to open stream: No such file or directory in /home/webs/affiliatelib2/CacheManager.php on line 130

Warning: fwrite(): supplied argument is not a valid stream resource in /home/webs/affiliatelib2/CacheManager.php on line 131

Warning: fclose(): supplied argument is not a valid stream resource in /home/webs/affiliatelib2/CacheManager.php on line 132

In vector calculus, the divergence theorem, also known as Gauss' theorem, Ostrogradsky's theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface.

More precisely, the divergence theorem states that the outward flux of a vector field through a surface is equal to the triple integral of the divergence on 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 physics, in particular in electrostatics and fluid dynamics.

Intuition The intuitive content is simple. 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 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

Suppose V is a subset of Rn (in the case of n = 3, V represents a volume in 3D space) which is compact space and has a piecewise smooth function boundary (topology). If F is a continuously differentiable vector field defined on a neighborhood of V, then we have

\iiint\limits_V\left(\nabla\cdot\mathbf{F}\right)dV=\iint\limits_{\part V} \mathbf{F} \cdot \mathbf{n}\, dS.

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. Here ∂V is quite generally the boundary of V oriented by outward-pointing Surface normal, and dS is shorthand for ndS, the outward pointing unit normal field of the boundary ∂V.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.

The divergence theorem is frequently applied in these following variants (cf. vector identities):

\iiint\limits_V\mathbf{F}\cdot \left(\nabla g\right) + g \left(\nabla\cdot \mathbf{F}\right)dV=\iint\limits_{\part V}g \mathbf{F}\cdot d\mathbf{S} (this is the basis for Green's identities, if \mathbf{F}=\nabla f),

\iiint\limits_V \nabla g dV=\iint\limits_{\part V} g d\mathbf{S},

\iiint\limits_V \mathbf{G}\cdot\left(\nabla\times\mathbf{F}\right) - \mathbf{F}\cdot \left( \nabla\times\mathbf{G}\right) dV = \iint\limits_{\part V}\left(\mathbf{F}\times\mathbf{G}\right)\cdot d\mathbf{S},

\iiint\limits_V \nabla\times\mathbf{F} dV = \iint\limits_{\part V}d\mathbf{S} \times\mathbf{F}.

Note that the divergence theorem is a special case of the more general Stokes theorem which generalizes the fundamental theorem of calculus.

Example Suppose we wish to evaluate

\iint\limits_S\mathbf{F}\cdot \mathbf{n}dS,

where S is the unit sphere defined by x^2+y^2+z^2=1 and F is the vector field

\mathbf{F} = 2 x\mathbf{i}+y^2\mathbf{j}+z^2\mathbf{k}.

The direct computation of this integral is quite difficult, but can be simplified using the divergence theorem:

\iint\limits_S\mathbf{F}\cdot \mathbf{n} dS=\iiint\limits_W\left(\nabla\cdot\mathbf{F}\right)dV=2\iiint\limits_W\left(1+y+z\right)dV = 2\iiint\limits_W dV + 2\iiint\limits_W y dV + 2\iiint\limits_W z dV.

Since the functions y and z are odd function on S (which is a symmetric set in respect to the coordinate planes), one has

\iiint\limits_W y dV = \iiint\limits_W z dV = 0.

Therefore, 2\iiint\limits_W\left(1+y+z\right)dV = 2\iiint\limits_W dV = \frac{8\pi}{3} because the unit ball W has volume 4π/3.

Applications Fluid dynamics Used to simplify the conservation of mass, momentum and energy equations.

Electrostatics Applied to an electrostatic field we get Gauss's law: the divergence is a constant times the volume charge density.

Gravity Applied to a gravity we get that the surface integral is -4πGravitational constant times the mass inside, regardless of how the mass is distributed, and regardless of any masses outside.

====Spherically symmetric mass distribution====

In the case of a spherically symmetric mass distribution we can conclude from this that the field strength at a distance r from the center is inward with a magnitude of G/r² times only the total mass within a smaller distance than r. All the mass at a greater distance than r from the center can be ignored.

For example, a hollow sphere does not produce any gravity inside. The gravitational field inside is the same as if the hollow sphere were not there (i.e. the field is that of any masses inside and outside the sphere only).

See also the shell theorem.

Cylindrically symmetric mass distribution In the case of an infinite cylindrically symmetric mass distribution we can conclude that the field strength at a distance r from the center is inward with a magnitude of 2G/r times the total mass per unit length at a smaller distance, regardless of any masses at a larger distance.

For example, an infinite hollow cylinder does not produce any gravity inside.

Bouguer plate We can conclude that for an infinite, flat plate (Bouguer plate) of thickness H gravity outside the plate is perpendicular to the plate, towards it, with magnitude 2πG times the mass per unit area, independent of the distance to the plate (see also Physical geodesy#Gravity anomalies).

More generally, for a mass distribution with the density depending on one Cartesian coordinate z only, gravity for any z is 2πG times the difference in mass per unit area on either side of this z value.

In particular, a combination of two equal parallel infinite plates does not produce any gravity inside.

History 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. Subsequently, variations on the Divergence theorem are called Gauss's Theorem, Green's theorem, and Ostrogradsky's theorem.

-------------------

This article was originally based on the GFDL article from PlanetMath at http://planetmath.org/encyclopedia/Divergence.html

Divergence theorem - Wikipedia, the free encyclopedia
In vector calculus, the divergence theorem, also known as Gauss's theorem (Carl Friedrich Gauss), Ostrogradsky's theorem (Mikhail Vasilievich Ostrogradsky), or Gauss-Ostrogradsky ...

Divergence Theorem
Divergence Theorem ... Stokes' Theorem Up: Several Variable Calculus Previous: Vector Calculus Formulas

10.3 The Divergence Theorem
The divergence theorem is the form of the fundamental theorem of calculus that applies when we integrate the divergence of a vector v over a region R of space.

Divergence Theorem -- from Wolfram MathWorld
A theorem in vector calculus more commonly known in older literature as Gauss's theorem. Let V be a region in space with boundary partialV. Then the volume integral of the ...

Divergence theorem - encyclopedia article - Citizendium
We are creating the world's most trusted encyclopedia and knowledge base. Once you join us and log in, you'll be able to edit this page instantly!

Divergence Theorem

Divergence - Wikipedia, the free encyclopedia
The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region is made precise by the divergence theorem.

Divergence theorem - Related Articles - Citizendium
We are creating the world's most trusted encyclopedia and knowledge base. Once you join us and log in, you'll be able to edit this page instantly!

Lecture 9: Divergence Theorem
Lecture 9: Divergence Theorem . Consider migrating wildebeest, with velocity vector field . 1 m . x . y . Zero influx (s-1) into blue triangle: 0 along hypotenuse

Divergence Theorem
math 220 supplemental notes 34. the divergence theorem and unified theory. diverence in three dimensions