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.

 

[blue_raver22]

Independent terms in the Magnetic Tensor refer to the components that are not dependent on each other. In the case of the magnetic gradient tensor, there are three magnetic field components and three baselines, which together form a 3x3 matrix. However, not all nine components are independent of each other. This means that some of the components can be calculated or derived from the others, making them redundant.

In this case, it has been determined through mathematical analysis that only five of the nine components are truly independent. This means that the other four can be calculated or derived from the five independent components. The five independent components are chosen in such a way that they provide the most accurate and complete representation of the magnetic gradient tensor.

The independence of these five components is due to the fact that they capture different aspects of the magnetic field and are not affected by the same factors. This allows them to provide unique information about the magnetic field and eliminate any redundancies. By using these independent components, we can simplify the analysis of the magnetic gradient tensor and make more accurate interpretations and predictions.