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Divergence theorem example question (Thomas' Calculus)

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Hey guys,
I have a general question about example 4 in section 16.8 of the book "Thomas' Calculus, Early Transcedentials". So far I understand the material given in the book without any problems but this particular example is a little bit unclear to me.

Homework Statement


Given a vector field [itex]\vec{F} = \frac{x\vec{i} + y\vec{j} + z\vec{k}}{ρ^3}[/itex], where [itex]ρ = \sqrt{x^2 + y^2 + z^2}[/itex], calculate the net flux of the field across the boundary of the region [itex]D: 0 < a^2 \leq x^2 + y^2 + z^2 \leq b^2[/itex], which appearently is the region bounded by two spheres of radii a and b respectively.


Homework Equations


The most relevant equation is obviously that of the divergence theorem:
Net flux [itex]\displaystyle \iint_{S} \vec{F}\cdot\vec{n}dσ = \iiint_{D}\nabla\cdot\vec{F} dV[/itex]
Where [itex]\vec{n}[/itex] is an outward unit normal vector to the surface S bounding D and [itex]div\vec{F} = \nabla\cdot\vec{F}[/itex] is the divergence of [itex]\vec{F}[/itex]

The Attempt at a Solution


I have confirmed the divergence of the field to be zero, as calculated in the solution of the example. So taking the triple integral of the divergence over the region D should also be zero,i.e:
[itex]\displaystyle \iiint_{D}\nabla\cdot\vec{F} dV = 0[/itex]
Okay. So far so good!

Next one can confirm this result using the surface integral version, considering the surface S consists of the surfaces of the two spheres bounding D. The example proves that the outward flux of any sphere is 4π, independent of the radius. In particular for the inner sphere:
[itex]\displaystyle \iint_{S1} \vec{F}\cdot\vec{n}dσ =\frac{1}{a^2}\iint_{S1}dσ = 4\pi [/itex]

Also since the two spheric surfaces have opposite unit normal vectors, the respective fluxes cancel out, giving yet again a total net flux of zero. Okay! I understand this.

Now lets consider an example of a region D bounded by only one surface S, described by a sphere with radius a.I.E. [itex]D: 0 \leq x^2 + y^2 + z^2 \leq a^2[/itex]. Then:
[itex]\displaystyle \iint_{S} \vec{F}\cdot\vec{n}dσ = 4\pi \neq \iiint_{D}\nabla\cdot\vec{F} dV = 0[/itex]

Now perhaps I'm missing the point or I still dont understand the theorem completely, but the definition "diverges" a little bit(no pun intended).
Can anyone please shed some light on this, its driving me insane!
Also I'm sorry if my question has been discussed already previously and my thread is redundant, or I posted in the wrong section. I'm new to this forum :)
Thanks in advance!
 
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Answers and Replies

  • #2
pasmith
Homework Helper
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Hey guys,
I have a general question about example 4 in section 16.8 of the book "Thomas' Calculus, Early Transcedentials". So far I understand the material given in the book without any problems but this particular example is a little bit unclear to me.

Homework Statement


Given a vector field [itex]\vec{F} = \frac{x\vec{i} + y\vec{j} + z\vec{k}}{ρ^3}[/itex], where [itex]ρ = \sqrt{x^2 + y^2 + z^2}[/itex]

[snip]

Now lets consider an example of a region D bounded by only one surface S, described by a sphere with radius a.I.E. [itex]D: 0 \leq x^2 + y^2 + z^2 \leq a^2[/itex]. Then:
[itex]\displaystyle \iint_{S} \vec{F}\cdot\vec{n}dσ = 4\pi \neq \iiint_{D}\nabla\cdot\vec{F} dV = 0[/itex]

Now perhaps I'm missing the point or I still dont understand the theorem completely, but the definition "diverges" a little bit(no pun intended).
Can anyone please shed some light on this, its driving me insane!
[itex](x,y,z)/\rho^3[/itex] is not defined at the origin, so in this case the divergence theorem does not apply to any volume which contains the origin.
 
  • #3
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Thanks for the fast response! So in other words one may calculate the outward flux as 4π, yet the net flux tends to 0 as ρ tends to 0? Hmm this wasn't clarified that well in the book :(
 

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