Timeline for Helmholtz theorem: “There is no function that has zero divergence and zero curl everywhere and goes to zero at infinity”. How do I know this?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| May 6, 2023 at 7:43 | comment | added | NotMe | Yes, $\vec u = 0$ everywhere satisfies the three conditions. However, the emphasise is put onto goes, which usually implies that it is non-zero somewhere. | |
| May 6, 2023 at 7:07 | history | edited | Qmechanic♦ | CC BY-SA 4.0 |
deleted 20 characters in body; edited tags
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| May 6, 2023 at 6:36 | answer | added | user365522 | timeline score: 2 | |
| May 5, 2023 at 22:52 | comment | added | nicoguaro | Does $\mathbf{u} = \mathbf{0}$ satisfy the conditions? Because $\nabla \times \mathbf{u} = \mathbf{0}$ and $\nabla \cdot \mathbf{u} = 0$ and $\lim_{\mathbf{x} \rightarrow 0} \mathbf{u}= 0$. | |
| May 5, 2023 at 22:33 | comment | added | HEMMI | I'm not good at English, but I thought that Griffith's English in the first quote might be misleading. I think it should be "there is no non zero function with zero ..." instead of "there is no function with zero ..."? | |
| May 5, 2023 at 22:28 | comment | added | hyportnex | yes, $\text{curl} \mathbf = 0$ is equivalent to that $\oint \mathbf u \cdot d\ell =0$ for any loop from which you can uniquely define $\phi(\mathcal P) =\int_{\mathcal P_0}^{\mathcal P} \mathbf u \cdot d\ell$ for some fixed $\mathcal P_0$ and variable $\mathcal P$, and finally $\frac{d\phi}{d\ell}=\text{grad} \phi =\mathbf u $ | |
| May 5, 2023 at 22:16 | comment | added | Helios | Though I have a question, does curl u = 0 enforces u = grad phi. Where phi is some scalar? | |
| May 5, 2023 at 22:11 | comment | added | Helios | That makes a lot of sense yeah, thank you. And by uniqueness theorem that it applies to all functions I am guessing. Thank you | |
| May 5, 2023 at 21:58 | comment | added | hyportnex | Does this help: If $\text{curl} \mathbf u = 0$ then $\mathbf u = \text{grad} \phi$ for some scalar $\phi$ and from $\text{div} \mathbf u = 0 $ you get $\text{div grad} \phi = \nabla^2 \phi =0$. | |
| S May 5, 2023 at 21:50 | review | First questions | |||
| May 5, 2023 at 22:54 | |||||
| S May 5, 2023 at 21:50 | history | asked | Helios | CC BY-SA 4.0 |