How did ostrogradsky prove the divergence theorem?

In summary, the divergence theorem, also known as Gauss's theorem, was first proved by Gauss and later by Ostrogradsky. However, it was actually discovered by Lagrange. There may be different variations of the proof for this theorem, but it is typically proven using the Fundamental Theorem of Calculus and repeated integration. It is suggested to search for the original articles at a university library, but be aware that Gauss had a unique way of writing mathematics.
  • #1
semidevil
157
2
so we know that the divergence theorm, was proved by Gauss, and also proved by ostrogradsky.

but infact, the divergence theorm was discovred by Lagrange...correct?

now, did these 3 guys prove it differently? I'm sure it couldn't be exactly the same way right? basically, I've been searching the internet all over, and can't find the differen't variations of the proof for the divergence theorem...

any help?
 
Last edited:
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  • #2
How about trying to search over the internet the original articles and then try to find them at the closest university library...?Pay attention with Gauss,he had a weird/funny way of writing mathematics...That's what i heard.

Daniel.
 
  • #3
thats an interesting question. this theorem follows so directly from the FTC plus repeated integration I cannot imagine any other way of proving it.

But whenever I say that, someone proves me wrong.
 

Related to How did ostrogradsky prove the divergence theorem?

1. How did Ostrogradsky prove the divergence theorem?

Ostrogradsky proved the divergence theorem by using the method of exhaustion, which involves dividing a region into smaller and smaller parts until the desired result is obtained. He also used the concept of infinitesimal calculus to derive the formula for the divergence of a vector field.

2. What is the significance of Ostrogradsky's proof of the divergence theorem?

Ostrogradsky's proof of the divergence theorem is significant because it provides a mathematical basis for understanding the flow of vector fields and their relationship to their boundaries. This theorem is also an important tool in many areas of physics, such as fluid mechanics and electromagnetism.

3. What are the main steps in Ostrogradsky's proof of the divergence theorem?

The main steps in Ostrogradsky's proof of the divergence theorem involve dividing a region into smaller parts, calculating the flux of the vector field through each part, and then taking the limit as the size of the parts approaches zero. This results in the formula for the divergence of the vector field, which relates the flow of the vector field to its sources and sinks.

4. How does the divergence theorem relate to other theorems in calculus?

The divergence theorem is closely related to other theorems in calculus, such as Green's theorem and Stokes' theorem. In fact, Green's theorem and Stokes' theorem can be derived from the divergence theorem by using the appropriate vector identities. This allows for a more comprehensive understanding of vector calculus and its applications.

5. Can the divergence theorem be applied to three-dimensional regions?

Yes, the divergence theorem can be applied to three-dimensional regions. In fact, Ostrogradsky's proof of the divergence theorem extends to any number of dimensions, making it a powerful tool for analyzing vector fields in higher dimensions. This allows for a deeper understanding of complex physical systems and their behavior.

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