# How do I visualize the in-homogeneous magnetic field?

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.

• 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 – Quantumness May 24 '18 at 2:34
• youtube.com/watch?v=muzGKFLNwJs From 46:00 – 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.