Brief introduction to tensors

In the note Magic Tensor Algebra was given a very good introduction to the mathematics of tensors. But, as it seems to me, this text is still somewhat difficult to understand. It is not completely clear what the tensor is and why it is needed at all.

Now I will try to give a very simple introduction to tensors. I do not pretend to mathematical rigor, so some terms may not be used quite correctly.

Where does the term tensor come from?

As I recall, the term tensor is derived from the Latin tensus or the English word tension - stress. The term arose in the process of understanding the next task. Let us be given some rigid body of arbitrary shape in three-dimensional space. Some forces are applied to different ends of the body. How to describe the resulting stresses in some section of this body? The answer to this problem is that the stresses are described by a tensor field. But to understand this answer, let's look at simpler tasks.

Zero-rank tensor

Let us be given in a three-dimensional space a homogeneous cube. Let's start to heat it from any side. Now let's fix some moment of time and try to describe the temperature values ​​at each point of the cube.
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Temperature is a scalar, we only need one number. We introduce an arbitrary coordinate system. Within this coordinate system, the temperature will be described as a scalar function of (x, y, z) .

And now let's take another coordinate system. What will change? And nothing! The temperature at each point in space remained the same scalar and did not change when the coordinate system was changed.

That's interesting! We have obtained a certain mathematical object, a scalar, which does not change when the coordinate system changes. We call it the zero rank tensor. Go ahead.

(Specification from the comments: the coordinates of the points will change, but the temperature at these points will not change due to the rotation of the coordinate system. It is the temperature that is the rank tensor (0,0))

First rank tensor

So, we have heated our homogeneous cube. Under the influence of the temperature of the molecules of any substance in it began to somehow move. Again, we fix some point in time and try to describe the velocity values ​​of the molecules at each point of the cube.

Speed ​​is a vector. We introduce an arbitrary coordinate system. Within this system, the speeds at each point in space will be described as vector functions of (x, y, z) . And now let's take another coordinate system? What will change? Let's reason.

The vector velocity field in the cube has not changed, it remains the same, we just took another ruler (another coordinate system) to measure the speeds. But the components of this vector have changed. Knowing the old and new coordinate system, the law of change of the vector components is easy to derive.

Thus, we obtained a mathematical object, a vector, which, again, does not change when the coordinate system changes, but its components change, and according to a predetermined law. This is a first rank tensor. Now the fun begins.

Second rank tensor

We heated our cube, the molecules began to move. But imagine now that our cube has ceased to be homogeneous. It is now porous, inside it consists of different channels with different orientations. The speed of movement of the molecule along the channel is much greater than the speed of movement across the channel. How do we describe such a heterogeneous environment?

Let's fix some moment of time, take one molecule with its velocity vector. The question is how this velocity vector will change in the next moment in time? If a molecule enters the channel and its velocity vector is directed along the channel, then the speed will not change, if the vector is directed across the channel, it will decrease several times, and if at an angle, then the velocity vector will generally change its direction.

It is very similar to the fact that at each point of the cube something is set that can rotate and scale the vectors. Yes, yes, this is a matrix! But not arbitrary, but special, which does not destroy the vector, but transforms.

Well, what will happen to our matrix if we take a different coordinate system, what will change? The configuration of channels in the cube remained the same, and this matrix should rotate the velocity vector in exactly the same way. Yes, the components of this matrix will change, but its very effect on vectors will remain the same.

Thus, we again have a mathematical object, a matrix of a special type, whose action on the vector does not depend on the change of the coordinate system, and its components are recalculated according to a certain law. We call it the second rank tensor.

So what is a tensor?

So, the tensor is a mathematical object, which as an object does not depend on the change of the coordinate system, but its components are transformed according to a certain mathematical law when the coordinate system changes. In three-dimensional space, a second-rank tensor is easiest to imagine as a matrix defined at each point in space, which describes the inhomogeneity of this space and acts on the incoming vector, changing its direction and scale.