Applying Divergence Theorem to Stokes' Theorem

If it is closed, then the triple integral will be zero. If not, the conclusion can be wrong.In summary, there is a discrepancy in using the divergence theorem and Stokes' theorem together, as the triple integral from Stokes' should be equal to the triple integral from the divergence theorem, but can result in a different value if the surface is not closed. The issue lies in the definition of the surface and whether it is considered closed or not.
  • #1
schaefera
208
0
Not really a homework problem, just me wondering about this: why is there a problem here?

Say you want to use the divergence theorem in conjunction with Stokes' theorem. So, from Stokes' you know: Line integral (F*T ds)= Surface integral (curl(F)*n)dS.

And you know that Surface integral(F*n)dS= Triple integral (div(F) dV))

But then, if you try to apply that to Stokes' you get: Triple integral (div(curl(F)) dV) which has to be 0, because div(curl(F))=0, right?

What's wrong with my reasoning?
 
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  • #2
I think you should check if one of the theorem states the surface as a "closed" one.
 

1. What is the difference between Divergence Theorem and Stokes' Theorem?

The Divergence Theorem states that the flux 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. Stokes' Theorem, on the other hand, relates the circulation of a vector field around a closed curve to the surface integral of the curl of that vector field over any surface bounded by the curve. In simpler terms, the Divergence Theorem deals with the relationship between a vector field and its divergence, while Stokes' Theorem deals with the relationship between a vector field and its curl.

2. When should I use Divergence Theorem and when should I use Stokes' Theorem?

Both theorems are used to evaluate integrals involving vector fields, but they are applicable in different situations. Divergence Theorem is used when the vector field is defined over a region in three-dimensional space, while Stokes' Theorem is used when the vector field is defined over a surface in three-dimensional space. In general, if the surface is closed, Divergence Theorem is used, and if the surface is open, Stokes' Theorem is used.

3. What is the relationship between Divergence Theorem and Stokes' Theorem?

Divergence Theorem and Stokes' Theorem are both fundamental theorems in vector calculus and are closely related. In fact, Stokes' Theorem is essentially a special case of Divergence Theorem, where the surface is a small loop around a point. This means that Stokes' Theorem can be derived from Divergence Theorem, and the two theorems are essentially equivalent.

4. How do I apply Divergence Theorem to Stokes' Theorem?

The process of applying Divergence Theorem to Stokes' Theorem involves using the concept of divergence and curl to convert one type of integral into the other. For example, if you have an integral involving a vector field and its divergence, you can use Divergence Theorem to convert it into an integral involving the same vector field and its curl. This can be helpful in simplifying calculations and solving problems in vector calculus.

5. What are some real-life applications of applying Divergence Theorem to Stokes' Theorem?

Both Divergence Theorem and Stokes' Theorem have numerous real-life applications in fields such as fluid mechanics, electromagnetism, and heat transfer. For example, in fluid mechanics, Divergence Theorem is used to calculate the flow of fluids through a closed surface, while Stokes' Theorem is used to analyze the circulation of fluids around a closed curve. In electromagnetism, these theorems are used to relate electric and magnetic fields to their sources and to calculate the work done by these fields. In heat transfer, they are used to study the flow of heat through different regions.

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