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I am just starting to become familiar with Tensors (most familiar with Moment of Inertia Tensor & Spring Constant Tensor), I am trying to understand the fundamental nature of them (as High-Level explanation as possible would be preferred)..

So a zeroth-order tensor is just a scalar.. A first-order tensor is a scalar and a direction (orientation)..

If a second-order tensor can be represented as a n x n matrix, it has scalar aspects, respective directions, but what is the third aspect that differentiates it from a 1st-order tensor?

I'm sure I am exposing some fundamental misunderstandings about Tensors, so any clarification is highly appreciated.

Shaman
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  • First-order tensors are simply vectors. They are objects that have a single index. Matrices have two indices, that is one simple way of seeing that they are fundamentally different objects. – Cyclone Sep 16 '17 at 20:15
  • Maybe helpful: https://math.stackexchange.com/questions/10282/an-introduction-to-tensors – Hans Lundmark Sep 16 '17 at 20:30
  • @HansLundmark very helpful.. thank you very much – Shaman Sep 16 '17 at 20:47

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Given that you seem to be interested in physics, I will give you an answer that is in accordance with how every physics class I have dealt with thinks about tensors.

Basically, the order of a tensor is how many indices you need to specifiy the object. For a scalar, there is no index. First order means I need one index to go across the vector. Second order means I have a table of values, rows and columns.

operatorerror
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  • Thank you very much for your response. That is enlightening, particularly in the context of Computer Science concepts. So what does the index imply about the object (or just in terms of the physical system at hand)? I know for, say, the Moment of Inertia Tensor, the indices represent the permutations of the Angular Velocity components and Angular Momentum components, but how can we generalize this notion? – Shaman Sep 16 '17 at 20:33