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I am a mere software engineer so my theoretical math knowledge is weak. I'm trying to get an intuitive understanding of tensors (in order to understand tensor voting). Here's how I'm thinking about it:

I have a point (a very small sphere) and for simplicity there's just two equal and opposite forces acting on it. Using vectors (first-order tensors) I could represent the sum of the force to show that it doesn't accelerate.

But let's say the sphere has some breaking strength, i.e. if the force on a point exceeds some threshold it will deform/break/etc. Using only vectors to sum the force on the sphere I get 0 net force, so that is not enough to tell me if the sphere will break or not.

So instead, I would represent the forces as tensors, and via tensor addition I could determine if the total force on the sphere exceeds the threshold.

Is that a sufficiently accurate way to think of a tensor? If not, could you explain why?

Tim Sweet
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  • I think you would want to sum the rank 3 tensors and if any entries have their absolute value above that point-stress-threshold for that point then you would see deformation there. – Urðr Nov 22 '17 at 20:25
  • I don't answer directly. You can view tensors as a generalization of matrices. In paticular, if you work with matrices, you already do tensor calculus. For example the product $MV=W$ of a matrix $M$( with entries $m_{i}^j$) by a vector $V$ (with coordinates $v_j$) is $m_{i}^jv_j=w_i$ (by indices **contraction" $i$...) – Jean Marie Nov 22 '17 at 20:40
  • see also (https://math.stackexchange.com/q/2316723). – Jean Marie Nov 22 '17 at 20:43
  • Thanks, I do understand the mathematical relationship between tensors, matrices, and vectors, I just don't understand what it all means in a deeper sense, so I'm trying to gain an intuitive understanding of it, just like the question you linked. It looks like that poster never really got the level of understanding he was hoping for, either. – Tim Sweet Nov 22 '17 at 20:48
  • All right. Besides, when you answer to somebody, please begin by arrowbas followed by his/her pseudo. – Jean Marie Nov 24 '17 at 08:32
  • No, you cannot represent forces as tensors, but the stress in a solid is a tensor. – user_of_math Dec 01 '17 at 14:33
  • @user_of_math is the distinction significant in my example? Couldn't we just assume each force on the sphere is acting on an area of 1 unit, thus the stress and force are equivalent? Or is there a distinction there which would make tensors easier to explain? – Tim Sweet Dec 02 '17 at 19:28

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