Timeline for Given a magnetic field how to find its vector potential? Is there an "inverse" curl operator?
Current License: CC BY-SA 4.0
5 events
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| Dec 18, 2022 at 5:46 | comment | added | nanoman | ... We'd get $\mathbf{A}'$ such that $\mathbf{A} = \boldsymbol{\nabla} \times \mathbf{A}'$. But, that is not what this answer actually seems to be doing. So, "the curl and divergence can be combined into a single operator that does have a unique inverse..." is confusing here because the answer purports to address OP's problem of solving $\boldsymbol{\nabla} \times \mathbf{A} = \mathbf{B}$ for $\mathbf{A}$, but doesn't mention $\boldsymbol{\nabla} \cdot \mathbf{A}$ at all. | |
| Dec 18, 2022 at 5:45 | comment | added | nanoman | ... The part that may be confusing is that OP's original problem sounds like the need is to reconstruct $\mathbf{A}$ (not $\mathbf{B}$) from its curl ($\boldsymbol{\nabla} \times \mathbf{A} = \mathbf{B}$). The remark "The curl on its own does not have a uniquely-defined inverse" seems to refer to this view of the problem: We know $\boldsymbol{\nabla} \times \mathbf{A}$ and we need $\mathbf{A}$. We could solve this by applying the Helmholtz decomposition to $\mathbf{A}$ (not $\mathbf{B}$), e.g., explicitly invoking Coulomb gauge $\boldsymbol{\nabla} \cdot \mathbf{A} = 0$. ... | |
| Dec 18, 2022 at 5:44 | comment | added | nanoman | This answer could use some clarification of how the mathematical Helmholtz decomposition maps to OP's physical problem. It appears that $\mathbf{F}$ (why isn't this vector bold when $\mathbf{A}$ is?) is intended to represent the magnetic field $\mathbf{B}$. Thus, the Helmholtz decomposition is really being used to decompose/reconstruct $\mathbf{B}$ (not $\mathbf{A}$) from its divergence ($\boldsymbol{\nabla} \cdot \mathbf{B} = 0$) and curl ($\boldsymbol{\nabla} \times \mathbf{B}$). This construction allows extracting a solution for $\mathbf{A}$ (which happens to be in Coulomb gauge). ... | |
| Dec 17, 2022 at 21:12 | vote | accept | lohey | ||
| Dec 17, 2022 at 20:49 | history | answered | Nullius in Verba | CC BY-SA 4.0 |