What are tensors? Why tensors are the main mathematical tool in physics?

Word “tensor” still remains very much for many physicists, and even more so for non-physicists, something little understandable, mathematical abstraction. And this despite the fact, that tensors themselves have been used in physics for over a century. What is a tensor? The answer to this question is extremely simple. – this collection of sets of numbers, which are associated with some physical object, isolated from the rest of the real world, each measurement procedure (that is, by comparing this entire object at once, or some of its individual properties, with selected scales) separately and by all such permissible measurement procedures at once. Tensors differ in the number of numbers in such sets and in the rules, which relate their values ​​in different coordinate systems.

These rules are simple, tensor classification too, but this simplicity requires clarification with illustrative examples.

Let's take for a start one-dimensional price space, used as an example when discussing the concept relativity. Let us have chosen as valid two currencies (two coordinate systems) ruble and dollar, and the physical object will be a bun.

First, simplest tensor, which appears in such a space – this scalar 1, assigned to such a property of physical objects, as quantity: 1 the bride. From the choice of the unit of measure for the price (ruble, li dollar) this property is independent, is an invariant and dimensionless number. Scalar is also called tensorohm zero rank. The scalar can take an arbitrary numeric value – 2, 3, 1.5 (brides). Note, that although scalars are dimensionless, but they have some rudimentary trace of dimension – rolls are different from sausages, eg, although in terms of price they are quite compatible. You can talk about the total price of rolls and sausages together. Ie. the difference between scalars is somewhat outside the scope of mathematics. The scalar is determined in space even before any measurement procedure is introduced., it arises as soon as we highlight individual parts in our world. But even after defining coordinate systems, it does not disappear.. This is the simplest set of numbers.. Component in set ” scalar” always alone. Its value is the same in all coordinate systems.. This fact, obviously, does not depend on the number of dimensions of space, because. the scalar itself from the measurement procedure, this dimension is defining, does not depend.

Next tensor, which we can immediately see, called vector, or the first rank tensor. This is nothing more than the price of a physical object (rolls in this case). Since our space is one-dimensional, only one property of an object in this space is described, then there will also be only one component in the vector. But! If a scalar has one component always, for space of any number of dimensions, then the vector has the number of components strictly equal to the number of dimensions. This is what is implied in the statement “first rank tensor“. When measuring, each scale corresponds to the selected object one dimensional component – number, indicating how many such scales are needed, to reproduce the object. Dimension of different components in general different and coincides with the name of the corresponding unit of measurement. In our particular case, it will be, eg, 25 rubles. Roll price in rubles, x^p=25 (rubles). And in dollars (in another coordinate system) it will be 1 dollars, x^d=1 (dollars). Note, that the transition coefficients between coordinate systems (currencies) two. From rubles to dollars, the ratio of the dollar to the ruble in a given market (is^d/is^p= 1/25 dollar / ruble) and vice versa (is^p/is^d= 25 ruble / dollar). Coordinate transformation coefficients are also dimensional, and have dimensions from both coordinate systems. Vector values ​​are converted from one coordinate system to another using the formula x^d=is^d/is^p•x^p. A completely natural formula. To get the values ​​of a vector component in a new coordinate system with respect to unit “dollars” you need to multiply the value of the vector in the old coordinate system with respect to one “ruble” of attitude new units to old. Notice, the dimensions are also transformed in this case! General rule – the components of such a vector are transformed during transitions between coordinate systems using the transformation matrix of the coordinates themselves (units in them selected), matrices of derivatives of new coordinates as functions of old. In our example, the matrix is ​​reduced to one number, but it is clear, what in the case of multiple units (multidimensional spaces) this will be a table of numbers (dimensional!).

It turns out, that in our one-dimensional space there also exist other tensors of the first rank, very similar to vector prices. They also have the same number of components., how many units of measurements in a given space. Therefore, they are also called vectors. But they express a completely different property of the object.! To distinguish between these two kinds of vectors, they are called contravariant vectors (such as price vector) and covariant vectors. These names mean “counter-transforming” and “co-transforming”. Easy to understand, what is connected with the formulas for the transformation of their components during transitions between coordinate systems. Now we will introduce a covariant vector for the bun, and you will see the difference. How many rolls can you buy per unit of measurement (in this case, prices – for one ruble or one dollar)? The question is quite meaningful, we often ask them. Exactly this object property, “fall in such and such quantity per unit of measurement” and expresses the covariant vector. For a loaf, it will be x_p=1/25 (1/ruble). Ie. on 1 a ruble can buy 1/25 of a loaf. note, the currency index is at the bottom and the dimension of the component of the covariant vector is inverse to the dimension of the corresponding unit. In a different coordinate system x_d=is^p/is^d•x_p. The components of the covariant vector are multiplied by the ratio old units to new.General rule, distinguishing a covariant vector from a contravariant – its components are transformed using the inverse coordinate transformation matrix, matrices of derivatives of old coordinates with respect to new.

Why do we need all these vectors?? In life, we add up prices, multiply, divide… Right, i have to enter (describe) tensor operations. Here's an example of multiplication, which has another special name, convolution. x^p•x_p= 1. What happened as a result of this operation, product of the price of a roll by the unit price of the same roll? Right, scalar, number of rolls, namely this one bun. And here's another ratio – from^p=x^p + and^p. What does it say? Product x has a certain price in rubles, product and another. The total price of the new product from, consisting of two products together, denote as from^p. What can you say about such from^p=x^p + and^d the sum? Or about such from^p=x^p + and_p? Or about such from_p=x^p + and^p? Stupidity, you can't fold it like that – rubles with dollars, or price with unit price. And you cannot add two prices to get the unit price. Adding prices always gives a price. So much for you general rule of tensor operations, which you may know as covariance requirement of the laws of physics. Addition, subtraction and equality can connect only tensors of the same structure in the same coordinate system. And this rule is completely natural, it simply spells out the common sense requirements described above.

Price space, which I chose for my examples, too simple, since one-dimensional and because of this introduce more complex tensors in it (second, and so on ranks) not easy enough (formally you can, but they don't make much sense). But it is very descriptive., understandable, operations in it are familiar to almost everyone. I want to emphasize one more important thing once again., which I tried to make clear to you, natural. Tensor, no matter how difficult it is at first, second and third look, there is always nothing else, as a numerical expression of some measured properties of some specific, selected physical object. Moreover, at this level (for a tensor of a given rank) exactly as many numbers, how many properties of an object are considered. And there are so many properties to consider, how many different independent units do you have. Better yet, express this thought in reverse. – for a complete description of an object, you must take as many independent units of measurement, how many independent properties does a given item have. In our particular case, we are interested in one property, price. So we have one unit of measurement..

Consider a more complex example, but still close to our direct experience. AND, of course, close to the theme of this site. Such an example can provide us with a 3-dimensional space.. At school, the first concept of tensor (true without mention, that we are talking about the tensor) we get by example velocity vectors. Point velocity vector (in the school course of physics and any solid) body has 3 Components, by the number of dimensions of space. He is usually depicted as an arrow., attached to the body and the radius vector in the figures. Need to clarify, that the concept of a radius vector is not an exact equivalent to the concept of a vector, because. the radius vector is associated with more than one point, but always with two. but, the concept of a vector at a given point is obtained as a result of the passage to the limit from the concept of a radius vector as the second point tends to the selected. AND, Besides, in Euclidean space, both concepts are often interchangeable, at least when depicted graphically. You can check, that the parallelogram rule for vector addition gives exactly the same result as explicitly (by component) their sum written in the standard tensor algebra. Addition of two velocity vectors (these are contravariant vectors): vⁱ=uⁱ + wⁱ , i=1,2,3. For three-dimensional space it is much easier to give examples of tensors and the following ranks. One of such important second rank tensors for the familiar Euclidean space is metric tensor  g_I, having in all orthogonal coordinate systems the diagonal form: g_I= 1 for i=k i = 0 at i≠k . This tensor is used to calculate the value of any contravariant vector, including, of course, and the velocity vector:  v² =∑g_I vⁱ v^k , where summation is performed over all values ​​of both indices, and size c (like v²) is a scalar. This formula writes down nothing more than the Pythagorean theorem in relation to the three-dimensional velocity vector.

My first example didn't let me show, what sets of numbers, resulting from measurements, may not constitute a tensor, and still remain meaningful precisely and only as an aggregate. The number of components, composing a meaningful geometric object, is also the same in different coordinate systems. But the law of their transformation during transitions between coordinate systems is more complicated, than tensor. An example is the coordinates themselves. They are always the same number., but the coordinates of one observer can be nonlinear functions of the coordinates of another. The second allows you to introduce and discuss other geometric objects., but I have an article, dedicated to just the most significant such object for physics — affine connection. Therefore, the, who wishes to deepen their understanding of this issue, better read this article.

Of course, tensors are also geometric objects, only some of their more special cases. Here I want to clarify for you, what exactly is the singling out of tensors of all such meaningful sets of measurements, geometric objects. Tensors are always associated with a specific object.. Geometric objects with transformation law, other than tensor are associated with variable objects, in one measurement procedure (coordinate system) they are talking about the same object, and in the other – about other. Tensors and operations with them provide a way, without thinking too much and always right, operate with the results of measuring the properties of selected objects. Of course, if we correctly understand the meaning of each tensor we use. But this is no longer a matter of mathematics., but the interpretation, applying mathematics to the real world.

Well, seems to be the general idea of ​​tensor and why their use is convenient, and most importantly, important and inevitable, i made it clear.

© Gavryusev V.G.
The materials published on the site can be used subject to the citation rules.