What are independent terms in Magnetic Tensor

[welshrich]

I am trying to understand the magnetic gradient tensor which has nine components. There are three magnetic field components, but there are also three baselines. These nine gradients are organised into a 3x3 matrix. I have read that only 5 of these terms are independent. What exactly does this mean? What makes them independent?

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

 

[welshrich]

No new findings. I think it has to do with symmetry but can't visualise it.

 

[Matterwave]

Probably if you put this thread in the physics section of the forums, you would have gotten better responses, since this problem doesn't really have to do with differential geometry. Really only 5 of the elements are independent because of the Maxwell's equations.

$$\nabla\times \vec{B}=\mu_0\left(\vec{J}+\epsilon_0\frac{\partial \vec{E}}{\partial t}\right)$$

This imposes 3 conditions on the 9 possible derivatives.

$$\nabla\cdot\vec{B}=0$$

This imposes 1 more condition on the 9 possible derivatives, leading to a total of 4 conditions on 9 numbers, leaving 5 numbers independent.