Alternate forms of Stokes' theorem? Are they correct? Are they named?

In summary, the conversation discusses various forms of Stokes's theorem and its applications in fluid dynamics. The speaker points out a missing minus sign in one of the equations and questions the correctness and individual names of the results. They also mention using dot products and divergence theorem to prove the identities and provide an example for the first identity.
  • #1
Hiero
322
68
TL;DR Summary
I have a few 3D formulas akin to Stokes theorem taught to undergraduates (not the more powerful diff forms version).
The last formula is what I was going for, since it arises as the momentum flux in fluid dynamics, but in the process I came across the rest of these formulas which I’m not sure about.

The second equation is missing a minus sign (I meant to put [dA X grad(f)]).
FFD5CDF9-E1D9-4085-A01D-0331B98D5720.jpeg
Are they correct? Do they have individual names? Or they’re all just roughly called ‘stokes theorems’?
 
Physics news on Phys.org
  • #3
If these results are correct then they can be established by dotting one side with an arbitrary constant vector field. The aim in case of a volume integral is then to write the integrand as a divergence, and for a surface integral to write the integrand as a curl and apply the divergence theorem or Stokes' Theorem respectively. Now manipulate the result to obtain a dot product of the arbitrary constant field with an integral. You can then 'cancel' the dot product. Setting [itex]d\mathbf{S} = \mathbf{n}\,dS[/itex] where [itex]\mathbf{n}[/itex] is the outward unit normal and [itex]d\mathbf{l} = \mathbf{t}\,d\ell[/itex] where [itex]\mathbf{t}[/itex] is the unit tangent and taking advantage of the cyclic nature of the scalar triple product is also useful.

For example, to prove the first identity: Let [itex]\mathbf{c}[/itex] be an arbitrary constant vector. Then [tex]
\nabla \cdot (\mathbf{f} \times \mathbf{c}) = \mathbf{c} \cdot (\nabla \times \mathbf{f}).[/tex] Now apply the divergence theorem: [tex]
\mathbf{c} \cdot \int_V \nabla \times \mathbf{f}\,dV = \int_{\partial V} (\mathbf{f} \times \mathbf{c}) \cdot \mathbf{n}\,dS = \mathbf{c} \cdot \int_{\partial V} \mathbf{n} \times \mathbf{f}\,dS.[/tex] Hence [tex]
\mathbf{c} \cdot \left(\int_V \nabla \times \mathbf{f}\,dV - \int_{\partial V} \mathbf{n} \times \mathbf{f}\,dS\right) = 0[/tex] and since this holds for all [itex]\mathbf{c}[/itex] it holds in particular for each of the cartesian standard basis vectors, and we must have [tex]\int_V \nabla \times \mathbf{f}\,dV = \int_{\partial V} \mathbf{n} \times \mathbf{f}\,dS.[/tex]
 
  • Like
Likes vanhees71

Related to Alternate forms of Stokes' theorem? Are they correct? Are they named?

1. What is the difference between the two forms of Stokes' theorem?

The two forms of Stokes' theorem are the classical form and the generalized form. The classical form applies to closed surfaces, while the generalized form applies to open surfaces.

2. Can both forms of Stokes' theorem be used interchangeably?

No, the two forms of Stokes' theorem have different applications and can only be used in specific situations. The classical form is used for closed surfaces, while the generalized form is used for open surfaces.

3. Are there any other names for the two forms of Stokes' theorem?

The classical form of Stokes' theorem is also known as the Kelvin-Stokes theorem, while the generalized form is also called the Cartan-Stokes theorem.

4. Can Stokes' theorem be applied to any type of surface?

No, Stokes' theorem can only be applied to smooth surfaces, meaning that the surface must have a well-defined tangent plane at every point.

5. How is Stokes' theorem related to other theorems in mathematics?

Stokes' theorem is a generalization of several other theorems in mathematics, including Green's theorem, Gauss's divergence theorem, and the fundamental theorem of calculus. It can also be seen as a higher-dimensional version of the fundamental theorem of line integrals.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Mechanical Engineering
Replies
4
Views
4K
Replies
4
Views
2K
  • Sci-Fi Writing and World Building
Replies
21
Views
1K
  • Mechanical Engineering
Replies
1
Views
3K
  • STEM Academic Advising
Replies
13
Views
2K
  • Quantum Physics
Replies
20
Views
3K
  • Quantum Physics
3
Replies
87
Views
5K
Back
Top