Extended divergence theorem

In summary, the extended divergence theorem allows for the calculation of outward flux of a singular vector field through a surface by enclosing it in another surface and looking at the inward flux. This can be useful when dealing with difficult integrals over arbitrary surfaces, as it can be reduced to an easier integral by using Stokes theorem.
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"Extended" divergence theorem

...which enables us to calculate the outward flux of a singular vector field through a surface S by enclosing it in some other arbitrary surface and looking at the inward flux instead.

Is there any other application of this apart from the special case when div(F)=0 you can, informally speaking, "ignore" the singularity?
 
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  • #2
Taking integrals over arbitrary surfaces is not easy. If you surround the singularity with a second surface that is easy, e.g. a small sphere then the region bounded by the two surfaces contains no singularity and is divergence free. Stokes theorem now says that the flux across the first surface is negative the flux over the second. This reduces a hard integral to an easy one.
 

What is the Extended Divergence Theorem?

The Extended Divergence Theorem is a mathematical concept that relates the surface integral of a vector field to the volume integral of its divergence. It is an extension of the Divergence Theorem, which only applies to closed surfaces.

What is the significance of the Extended Divergence Theorem?

The Extended Divergence Theorem is significant because it allows for the calculation of flux through open surfaces, which cannot be done using the standard Divergence Theorem. It also has important applications in various fields of science, such as fluid mechanics and electromagnetism.

How is the Extended Divergence Theorem derived?

The Extended Divergence Theorem is derived from the fundamental theorem of calculus, which states that the integral of a derivative over a region is equal to the difference of the function evaluated at the boundaries of that region. By applying this principle to vector fields and using the Divergence Theorem, the Extended Divergence Theorem can be obtained.

What are the conditions for using the Extended Divergence Theorem?

In order to use the Extended Divergence Theorem, the vector field must be differentiable and the region must be bounded by a smooth surface. Additionally, the surface must be oriented such that the outward normal vector points away from the region.

How is the Extended Divergence Theorem applied in real-world problems?

The Extended Divergence Theorem has a wide range of applications in physics, engineering, and other scientific fields. For example, it can be used to calculate the flow rate of a fluid through a given surface or to determine the electric field produced by a charged object. It is also used in the development of numerical methods for solving differential equations.

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