In a Magnetic field, there are 4 terms (divergence, gradient, curvature and twist). I understand the divergence and gradient terms but why is the curvature term represented with $\frac{\partial B_x}{\partial z}$ & $\frac{\partial B_y}{\partial z}$ while the twist is represented with $\frac{\partial B_x}{\partial y}$ & $\frac{\partial B_y}{\partial x}$?

I can't visualize the curvature and twist terms and some visual aid will be appreciated.

  • $\begingroup$ Could you explain or link some source that uses this "curvature" and "twist?" Generally, $\vec{B}$ refers to the magnetic field, so what you wrote down would be an entity not the same as the magnetic field, of which I can not find any information about $\endgroup$ – Quantumness May 24 '18 at 2:34
  • $\begingroup$ youtube.com/watch?v=muzGKFLNwJs From 46:00 $\endgroup$ – newbie125 May 24 '18 at 9:59

The commented video refers to the tensor quantity $\nabla\mathbf{B}$. I do not know if there is some convention used particularly for plasma physics; however, from a more mathematical point of view written in index form this is $$\nabla\mathbf{B}=\frac{\partial B^\alpha}{\partial x^\beta}g^{\beta \gamma}$$ From this the four terms are seen to be the terms where $c$ is one of the integers $1,2,3$

$\alpha=\beta\implies\nabla\cdot\mathbf{B}$ for divergence,

$\alpha=c\implies \partial_\beta B^c$ for gradient,

$\beta=c$ with $\alpha \neq\beta \implies\partial_cB^\alpha$ for curvature,

and $\epsilon^{ijk}\partial_jB^k= \nabla\times\mathbf{B}$ for shear (or twist)

The video explains curvature as how the fields "circle" about a point, and shear as how some component changes following some change in location.

This is where I may have misinterpreted or the professor may have made an error; the visualizations for the curvature and shear in math have switched formulas (i.e. "curvature" is curl), otherwise the professor's explanation is clear.

These links may be more helpful for the actual physics: http://fusionwiki.ciemat.es/wiki/Magnetic_curvature



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