What are independent terms in Magnetic Tensor

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Discussion Overview

The discussion centers on the magnetic gradient tensor, specifically its nine components and the assertion that only five of these terms are independent. Participants explore the implications of this independence and the underlying reasons, including potential connections to symmetry and Maxwell's equations.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant seeks clarification on the concept of independent terms within the magnetic gradient tensor and questions what makes them independent.
  • Another participant suggests that the independence may relate to symmetry but admits difficulty in visualizing this concept.
  • A different participant proposes that the independence of the five terms arises from conditions imposed by Maxwell's equations, citing specific equations that lead to constraints on the nine components.
  • This participant notes that four conditions result from Maxwell's equations, leaving five independent components.

Areas of Agreement / Disagreement

Participants express differing views on the nature of independence in the magnetic gradient tensor, with some suggesting symmetry as a factor and others emphasizing the role of Maxwell's equations. The discussion remains unresolved regarding a unified understanding of these concepts.

Contextual Notes

Participants have not fully explored the implications of symmetry or the specific conditions derived from Maxwell's equations, leaving some assumptions and definitions unaddressed.

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