Physical meaning of a tensor

 

 

 

 

 

 

Question about tensors

 

Hmm… What is a tensor..? What’s the difference between a tensor and a matrix..?

 

Did you call me?

 

Ah, Mr. Penguin. Nope.. I didn’t call you..

 

It seems like you are studying about tensors.

You are confusing a tensor with a matrix, aren’t you?

 

Actually… I am..

 

You don’t want to get the academic definition of a tensor but you’d like to get the image of a tensor, right?

 

Exactly!

 

If so, I think that we should think about the physical meaning of a tensor and that is the theme for today.

“What is the physical meaning of a tensor?”

First of all, let me ask you a question. Please give me an example of something that is NOT a tensor?

 

Well… I have no idea..

 

Characteristic of tensors (vectors)

 

That’s no problem. I’ll give you the example later.

I guess that you can give me an example of a tensor, can’t you?

 

According to a text book about tensors, a vector is one of tensors.

 

Good. Then, will you tell me the characters of a vector?

 

Well… A vector is a concept that has both a magnitude and a direction. Am I right?

 

Right. However, you are missing another important character of a vector.

The character is so obvious that it may be difficult to understand it. Please listen to what I’m going to say carefully.

A vector itself will not change even if the coordinate system changes into another one.

We can say the same thing to a tensor.

 

Well… Could you give me an example to help me understand it clearly?

 

Yes I can. As you know, a velocity is one of vectors.

There are two people in the picture below. Do you agree that the two people are looking at the same velocity vector?

 

 

Well… I don’t know what you are saying well..

 

I’m just saying a very simple thing. Okay, let me change my question.

Do you think the situation in the picture below happens?

 

 

No. The situation will never happen.

The situation is like someone says that the ball goes east but another person says that the ball goes south with a different speed. If this happened, the world would be messy..

 

You are right.  A vector doesn’t depend on who looks at. That is one of important characters of a vector.

 

 

Hmm.. but if the coordinate system changes, the components of a vector will change accordingly. Am I wrong?

 

 

It is no doubt that the components of the vector changed. However, has the vector itself changed?

 

No… Both the magnitude and the direction of the vector have not changed at all.

 

Right. A vector itself will not change even if the coordinate system changes into another one.

We can say the same thing to a tensor. A tensor itself will not change even if the coordinate system changes into another one.

 

Okay but.. I am not sure if I get it right…

 

What is NOT a tensor?

 

Don’t be worried. I’ll guide you. I think you should know something that is NOT a vector.

I prepared a matrix below for you. The matrix expresses the numbers of dogs and cats that my neighbor Michael keeps.

 

 

It looks like a vector… but you said that you prepared something that is NOT a vector..

 

It can NOT be a vector. Let’s try to change the coordinate system into another one. Please take a look at the picture below.

 

 

Wow.. the directions of the arrows are not the same…

 

Right. We can say that an arrow itself changes in response to a coordinate transformation and which means that the matrix above has a different concept of a vector and a tensor.

 

I feel like I understood the difference between a tensor and a matrix.

 

That’s good. Let’s call it a day.