# Divergence theorem example question (Thomas' Calculus)

• LastLight
In summary, the conversation discusses a question about example 4 in section 16.8 of the book "Thomas' Calculus, Early Transcedentials". It involves using the divergence theorem to calculate the net flux of a vector field across the boundary of a given region. The conversation goes into detail about the relevant equations and a specific example, with the conclusion that the theorem does not apply to any volume containing the origin.
LastLight
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 $\vec{F} = \frac{x\vec{i} + y\vec{j} + z\vec{k}}{ρ^3}$, where $ρ = \sqrt{x^2 + y^2 + z^2}$, calculate the net flux of the field across the boundary of the region $D: 0 < a^2 \leq x^2 + y^2 + z^2 \leq b^2$, 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 $\displaystyle \iint_{S} \vec{F}\cdot\vec{n}dσ = \iiint_{D}\nabla\cdot\vec{F} dV$
Where $\vec{n}$ is an outward unit normal vector to the surface S bounding D and $div\vec{F} = \nabla\cdot\vec{F}$ is the divergence of $\vec{F}$

## 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:
$\displaystyle \iiint_{D}\nabla\cdot\vec{F} dV = 0$
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:
$\displaystyle \iint_{S1} \vec{F}\cdot\vec{n}dσ =\frac{1}{a^2}\iint_{S1}dσ = 4\pi$

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

Now perhaps I'm missing the point or I still don't 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 :)

Last edited:
LastLight said:
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 $\vec{F} = \frac{x\vec{i} + y\vec{j} + z\vec{k}}{ρ^3}$, where $ρ = \sqrt{x^2 + y^2 + z^2}$

[snip]

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

Now perhaps I'm missing the point or I still don't 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!

$(x,y,z)/\rho^3$ is not defined at the origin, so in this case the divergence theorem does not apply to any volume which contains the origin.

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 :(

## 1. What is the Divergence Theorem in Thomas' Calculus?

The Divergence Theorem, also known as Gauss's Theorem, is a fundamental concept in vector calculus that relates the flux of a vector field through a closed surface to the divergence of the field within the region bounded by the surface. In simpler terms, it states that the net flow of a vector field through a closed surface is equal to the amount of "source" or "sink" within that surface.

## 2. How is the Divergence Theorem used in real-life applications?

The Divergence Theorem has many practical applications in physics and engineering, particularly in fluid mechanics and electromagnetism. For example, it can be used to calculate the flow of fluid through a pipe, or the electric flux through a closed surface surrounding a charged object.

## 3. Can you provide an example question using the Divergence Theorem?

One example question could be: "Find the flux of the vector field F(x,y,z) = through the surface S: x^2 + y^2 + z^2 = 4, z ≥ 0, oriented upward, using the Divergence Theorem."

## 4. How is the Divergence Theorem related to other theorems in calculus?

The Divergence Theorem is closely related to the Fundamental Theorem of Calculus, which states that the integral of a function over an interval can be calculated by evaluating its antiderivative at the endpoints of the interval. Additionally, it is a special case of the more general Stokes' Theorem, which relates the flux of a vector field through a surface to the line integral of the field around the boundary of the surface.

## 5. Are there any limitations to using the Divergence Theorem?

Yes, the Divergence Theorem can only be applied to smooth vector fields and surfaces. It also requires the surface to be closed, meaning it forms a complete boundary around a three-dimensional region. Additionally, the surface must have a consistent orientation, meaning all normal vectors point in the same direction. Violating any of these conditions can lead to incorrect results when using the Divergence Theorem.

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