Divergence theorem on non compact sets of R3

In summary, the divergence theorem states that the integral of the divergence of a continuously differentiable vector field over a compact subset of \mathbb{R}^3 with a piecewise smooth boundary surface S is equal to the surface integral of the vector field dotted with the outward unit normal vector over the boundary surface S. This theorem is often used to compute the electric field of a non-bound set, such as an infinite cylinder, by showing that the integral over any surface closing the area goes to zero as the closing surface is placed further away. In some cases, additional requirements on the integrand may be necessary, such as vanishing sufficiently fast. There are also cases where the volume is not infinite and symmetry may allow for reduction of the
  • #1
Lebesgue
So my question here is: the divergence theorem literally states that
Let [itex]\Omega[/itex] be a compact subset of [itex] \mathbb{R}^3 [/itex] with a piecewise smooth boundary surface [itex]S[/itex]. Let [itex]\vec{F}: D \mapsto \mathbb{R}^3[/itex] a continously differentiable vector field defined on a neighborhood D of [itex]\Omega[/itex].
Then:
[itex]\int_{\Omega} \nabla \cdot \vec{F} dxdydz = \oint_S \vec{F} \cdot \vec{n} dS [/itex]

My problem here is: why people (and with which argument) use this divergence or Gauss theorem to compute the electric field of some NOT bound set (for example, the typical infinite cylinder) of surface charge.
 
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  • #2
Doing so is fine if you can show that the integral over any surface closing the area goes to zero when that closing surface is placed further and further away. This may put additional requirements on the integrand, such as vanishing sufficiently fast.

There are also cases where the volume is not infinite although you are studying an infinite setup, but, for example, the end caps of thr cylinder do not contribute. In other cases, symmetry may reduce the dimensionality of the problem and it is sufficient to consider it in one dimension less where the volume is bounded.
 

Related to Divergence theorem on non compact sets of R3

1. What is the Divergence Theorem on non compact sets of R3?

The Divergence Theorem, also known as Gauss's Theorem, is a fundamental theorem in vector calculus that relates the surface integral of a vector field to the triple integral of its divergence over a region in three-dimensional space. When applied to non compact sets in R3, the theorem allows for the calculation of flux through open surfaces or surfaces that extend to infinity.

2. How is the Divergence Theorem applied to non compact sets of R3?

To apply the Divergence Theorem to non compact sets in R3, the region of interest must be bounded by a surface that extends to infinity. The surface must also be piecewise smooth and orientable, meaning that a consistent outward normal vector can be defined at each point on the surface. The triple integral of the divergence of the vector field over the region is then equal to the surface integral of the vector field over the boundary of the region.

3. What is the significance of the Divergence Theorem on non compact sets of R3?

The Divergence Theorem is significant in that it provides a powerful tool for solving physical problems involving vector fields, such as fluid flow and electric field calculations. By extending the theorem to non compact sets in R3, it allows for the calculation of flux through surfaces that would otherwise be difficult to integrate over.

4. What are some real-world applications of the Divergence Theorem on non compact sets of R3?

The Divergence Theorem has many applications in physics and engineering. For example, it can be used to calculate the net flow of a fluid through an open surface, the total electric charge enclosed by a surface, or the rate of heat transfer through a solid object. It is also used in the study of fluid dynamics, electromagnetism, and heat transfer.

5. Are there any limitations to the Divergence Theorem on non compact sets of R3?

While the Divergence Theorem is a powerful tool, it does have some limitations. It can only be applied to regions that are bounded by a surface that extends to infinity, and the surface must be piecewise smooth and orientable. Additionally, the vector field must be well-behaved over the region of interest. If these conditions are not met, the Divergence Theorem cannot be used to solve the problem.

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