Proofs of Stokes Theorem without Differential Forms

In summary, the speaker is looking for a rigorous proof of Stokes Theorem in three dimensions that does not use Differential Forms. They do not plan on using the higher dimensional version and find it a waste of time to learn Differential Forms. They provide a reference for a proof of Stokes Theorem using line integrals.
  • #1
Crek
66
11
Hello, does anyone have reference to(or care to write out) fully rigorous proof of Stokes theorem which does not reference Differential Forms? I'm reviewing some physics stuff and I want to relearn it.

I honestly will never use the higher dimensional version but I still want to see a full proof in the 3d case - my analysis texts use differential forms and my calculus books don't provide a real proof. I could learn differential forms but I will not use it and I will then forget, making it kind of a waste of time.
 
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  • #2
Check this reference:

https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/4.-triple-integrals-and-surface-integrals-in-3-space/part-c-line-integrals-and-stokes-theorem/session-92-proof-of-stokes-theorem/MIT18_02SC_MNotes_v13.3.pdf
 

1. What is the concept behind Stokes Theorem?

Stokes Theorem is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field along the boundary curve of the surface. It is a generalization of the fundamental theorem of calculus and is used to solve many problems in physics and engineering.

2. What are differential forms and how are they used in Stokes Theorem?

Differential forms are mathematical objects that are used to integrate vector fields over surfaces or volumes. They are a generalization of the concept of a function and are used to represent the flux of a vector field through a surface or the flow of a vector field through a volume. In Stokes Theorem, they are used to express the relationship between the surface and line integrals.

3. Can Stokes Theorem be proved without using differential forms?

Yes, there are alternative proofs of Stokes Theorem that do not involve the use of differential forms. These proofs often use the concept of flux or flow, and may require additional assumptions or techniques. However, the use of differential forms provides a more elegant and general proof of the theorem.

4. What are the advantages of using differential forms in Stokes Theorem?

The use of differential forms in Stokes Theorem allows for a more concise and general proof, as well as a deeper understanding of the relationship between the surface and line integrals. It also provides a framework for extending the theorem to higher dimensions and more complex surfaces.

5. Are there any applications of Stokes Theorem in real-world problems?

Yes, Stokes Theorem is widely used in various fields of physics and engineering to solve problems involving vector fields, such as fluid flow, electromagnetic fields, and heat transfer. It is also used in mathematical modeling and computer simulations to analyze complex systems.

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