A question regarding the definition of a tensor

[MicaGlom]

Hello,

I have recently started reading some notes on introduction to tensors, trying to get more familiar with this mathematical object. I have two questions I can't seem to answer myself:

1. A tensor is roughly defined in the text as a collection of quantities associated with a point in space, which transform according to an unchanging rule. What is meant by an unchanging rule? what exactly is NOT changing?
The following is how I answered to myself: an unchanging rule is a rule according to which the "collection of quantities" is transformed between coordinate systems, without changing the "collection of quantities" or the way it may be interpreted in each coordinates system. Am I right?

2. One line in the text states that "while every rank-0 tensor is a scalar, not every scalar is a rank-0 tensor". temperature is a clear example of a scalar quantity that can be considered a rank-0 tensor, but I could not think of any example for a scalar that is NOT a rank-0 tensor. Could someone please provide one?

Many thanks!

 

[Studiot]

1)

Yes you have the idea but the change of coordinate system cannot be totally arbitrary. It has to be chosen from any of the 'proper rotations' or from scalings. Reflection is also called an 'improper rotation' and is disallowed.

2)

Tensors obey the rules of linear algebra (plus some other rules of their own) so for, instance you can add two tensors in only one (linear) way

R+S = S+R = T

This is also true of some single quantity entities such as energy or mass.

So 4kg + 2 kg is 6 kg however you add them up.

Electrical resistance, however is a single quantity entity that cannot be handled in this way because adding two resistors in parallel yields a different result from adding them in series.

 

[MicaGlom]

Studiot,

Thank you very much for your response, It certainly helped me out.

 

[Studiot]

I should, perhaps, point out that there is something called the conductivity tensor in electromagnetic field theory. In an isotropic medium this tensor reduces to a single value - zero rank tensor.

This highlights a major difference between electrical resistance and resistivity.

go well