A question that uses divergence thm

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In summary, the conversation discusses the use of the divergence theorem to show that the volume enclosed by a closed surface S is equal to 1/3 times the integral of the position vector dotted with the area element, with the use of a simplified form of the divergence theorem. It is clarified that the position vector represents a vector field and that its divergence is equal to 3.
  • #1
Gregg
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Homework Statement

show that the volume enclosed by a closed surface S is given by

## \frac{1}{3} \int_{S} \vec{x} \cdot d\vec{A} ##

Homework Equations

divergence theorem

The Attempt at a Solution



using divergence theorem

I get that ##V =\frac{1}{3} \int_{V} \nabla \cdot \vec{x} dV ##

but I want

## V = \int_{V} dV ## don't I? how does it simplify?
 
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  • #2
Unless

## \vec{x} = (x,y,z) ##

Then:

## V = \frac{1}{3} \int_{V} (1+1+1) dV = \int_{V}dV ##

It's because x is just the position vector whose divergence is 3 isn't it? i.e. x is not a vector field?
 
  • #3
Sure, that's it. But the position vector (x,y,z) IS a vector field. It's just a particular one.
 

1. What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem, is a fundamental result in vector calculus that relates the flux of a vector field through a closed surface to the divergence of the field inside the surface.

2. How is the divergence theorem used in physics and engineering?

The divergence theorem is used in various applications, such as in electromagnetism to calculate electric and magnetic fields, in fluid mechanics to analyze fluid flow, and in heat transfer to calculate heat flux.

3. What is the significance of the divergence theorem in mathematics?

The divergence theorem is an important tool in vector calculus and is used to simplify calculations involving flux and divergence. It also helps in solving various boundary value problems in physics and engineering.

4. What are the conditions for the divergence theorem to hold?

The divergence theorem holds true for smooth vector fields and closed surfaces. The surface must be oriented in a consistent manner and the vector field must be continuous and differentiable inside and on the boundary of the surface.

5. Can the divergence theorem be extended to higher dimensions?

Yes, the divergence theorem can be extended to higher dimensions through the use of the generalized Stokes' theorem, which relates the integral of a differential form over a manifold to the integral of its exterior derivative over the boundary of the manifold.

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