# Divergence Theorem Question (Gauss' Law?)

• vector013
In summary, the problem asks you to take a ball of radius r centered at the origin, apply divergence theorem, and let the radius tend to infinity. However, you don't seem to be able to do this using divergence theorem, so you should instead use spherical coordinates.
vector013
If F(x,y,z) is continuous and
for all (x,y,z), show that R3
dot F dV = 0

I have been working on this problem all day, and I'm honestly not sure how to proceed. The hint given on this problem is, "Take Br to be a ball of radius r centered at the origin, apply divergence theorem, and let the radius tend to infinity." I tried letting F = 1/((x2 +y2+z2)(3/2))+1), and taking the divergence of that, but it didn't really seem to get me anywhere. If anyone has any suggestions for at least how to set up this proof, I would really appreciate it.

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Well, from the title it seems that you know you should use divergence theorem. So why don't you?

Since your region of integration is a ball and the integrand involves $x^2+ y^2+ z^2$, I would recommend changing to spherical coordinates to do that integration over the surface of the ball.

Spherical was my thought too. I guess what's been confusing me is that the vector field isn't explicitly given; there's just the inequality which indicates it exists at (0,0,0). My thought was to use Gauss' Theorem for Divergence in Spherical Coordinates: div F = 1/p2*d/dp*(p2Fp)+1/(psin(phi))*d/dphi*(sin(phi)Fphi + 1/(psin(phi))*dFtheta/dtheta. Since you have 1/p terms, div F would go to zero as the radius, p, goes to infinity. Therefore, the integral of 0 is 0.

vector013 said:
Spherical was my thought too. I guess what's been confusing me is that the vector field isn't explicitly given; there's just the inequality which indicates it exists at (0,0,0). My thought was to use Gauss' Theorem for Divergence in Spherical Coordinates: div F = 1/p2*d/dp*(p2Fp)+1/(psin(phi))*d/dphi*(sin(phi)Fphi + 1/(psin(phi))*dFtheta/dtheta. Since you have 1/p terms, div F would go to zero as the radius, p, goes to infinity. Therefore, the integral of 0 is 0.
I don't see any application of divergence theorem in your explanation. This is how it should be:
## \int_V \nabla\cdot \vec F dV=\int_{\partial V} \vec F \cdot \hat n dA ##
Where ## \partial V ## is the boundary of V. Now because in the surface integral, F is evaluated only at the boundary and the boundary is at infinity and we know that F is smaller than a function which goes to zero at infinity, the integral should be zero.

Oh okay, that makes sense. I didn't even think of it that way. Thank you so much for the suggestion!

## 1. What is the Divergence Theorem and why is it important in science?

The Divergence Theorem, also known as Gauss' law, is a fundamental principle in mathematics and physics. It states that the flux (flow) of a vector field through a closed surface is equal to the volume integral of the divergence of that vector field over the region enclosed by the surface. In simpler terms, it relates the behavior of a vector field inside a region to its behavior on the boundary of that region. This theorem is important in many areas of science, including electromagnetism, fluid dynamics, and thermodynamics.

## 2. How is the Divergence Theorem used in electromagnetism?

In electromagnetism, the Divergence Theorem is used to calculate the electric or magnetic flux through a closed surface. The flux is a measure of how many electric or magnetic field lines pass through a given area. The Divergence Theorem allows us to simplify the calculation of flux by relating it to the behavior of the electric or magnetic field inside the closed surface. This makes it a powerful tool for solving problems in electromagnetism, such as calculating the electric field due to a charged particle or a current-carrying wire.

## 3. Can the Divergence Theorem be applied to any vector field?

Yes, the Divergence Theorem can be applied to any vector field, as long as the field is well-behaved and the surface is closed. This means that the field is continuous and differentiable everywhere inside the region and the surface is a closed, smooth surface with no holes or edges. In practical applications, the vector field is often a physical quantity, such as velocity, temperature, or electric field, and the surface is a physical boundary, such as a container or a closed surface surrounding a charged object.

## 4. What is the difference between the Divergence Theorem and Gauss' Law?

The Divergence Theorem and Gauss' Law are essentially the same thing, but they are used in different contexts. The Divergence Theorem is a general mathematical principle that applies to any vector field, while Gauss' Law specifically refers to the behavior of the electric field in the presence of electric charges. In this context, the Divergence Theorem is often referred to as Gauss' Law in integral form, while the differential form of Gauss' Law is equivalent to one of the Maxwell's equations in electromagnetism.

## 5. How is the Divergence Theorem used in fluid dynamics?

In fluid dynamics, the Divergence Theorem is used to relate the behavior of a fluid inside a region to its behavior on the boundary of that region. This allows us to simplify the calculation of fluid flow, such as the rate of mass or energy flow, by considering only the behavior of the fluid at the boundaries. The Divergence Theorem is also a key component in the Navier-Stokes equations, which describe the motion of viscous fluids.

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