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

 

[pmacias]

Hello,

Thank you for your questions regarding the definition of a tensor. I am happy to help clarify these concepts for you.

1. When we say that a tensor transforms according to an unchanging rule, it means that the mathematical relationship between the quantities in the tensor remains the same, regardless of the coordinate system used. This means that the values of the quantities themselves may change, but the way they are related to each other remains constant. For example, if we have a tensor representing the stress at a point in a material, its components may change when we switch from Cartesian coordinates to polar coordinates, but the way they are related to each other through the mathematical equations governing stress will remain the same.

Your understanding of an unchanging rule is correct. The rule refers to the mathematical relationship between the quantities in the tensor, not the quantities themselves.

2. A scalar is a quantity that has only magnitude, and no direction. Examples of scalars include temperature, mass, and time. A rank-0 tensor is a tensor with no indices, meaning it has only one component. So, a scalar can be considered a rank-0 tensor because it has only one component. However, not every scalar can be considered a rank-0 tensor because it must also transform according to the rules of tensor transformation. For example, if we have a scalar quantity that behaves differently under coordinate transformations, it cannot be considered a rank-0 tensor. One example of this could be electric charge, as it changes sign under certain coordinate transformations.

I hope this helps clarify the concepts of a tensor and its relationship to scalars. Let me know if you have any further questions.

Best,