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In some differential equations from phyiscs, e.g. in elastodynamics, terms with the curl of the curl of a vector field appear. For example

$(\lambda + 2 \mu) \nabla(\nabla\cdot \mathbf{x}) - \mu \nabla\times\nabla \times \mathbf{x} = \rho \ddot{\mathbf{x}}$.

Is there a way to visualize or have an intuitive/physical understanding of what is represented by the curl of a curl of a vector field.

DiggingDeep
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  • This could be helpful. Not sure what can be said which you do not already know: a non zero vector field $F$ is of the form $F=\nabla\times\nabla\times A$ if and only if it is divergence free and has an "anti-curl" $G$ that is divergence free but not rotation free: $F=\nabla \times G,,G=\nabla\times A,.$ A few simple examples of vector fields $A$ that have non zero $\nabla\times\nabla\times A$ are – Kurt G. Jun 30 '23 at 12:07
  • $$ \begin{pmatrix}y^2\0\0\end{pmatrix},,\begin{pmatrix}z^2\0\0\end{pmatrix},,\begin{pmatrix}0\-2xy\0\end{pmatrix},,\begin{pmatrix}0\0\-2xz\end{pmatrix},. $$ They should not be that hard to visualize. In all those cases $\nabla\times\nabla \times A$ is $(-2,0,0),.$ – Kurt G. Jun 30 '23 at 12:07
  • Thanks for the input. it gives some ideas to build upon, even though I don't quite 'see' it yet:D. – DiggingDeep Jul 03 '23 at 10:23
  • I think you should first let it sink in why the following examples have / have not curl (vorticity). This can be quite counterintuitive for the parallel flow with shear (one of my favourites). It is simply $A=(y,0,0)$ having curl $(0,0,-1),.$ After that you can go one level higher. Also: a relatively simple anti-curl formula you can find here. – Kurt G. Jul 03 '23 at 10:29

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