Rotor (mathematics)

Rotor, or vortex- vector differential operator over a vector field.

Designated

(in Russian-language literature) or

(in English literature),

and also as a vector multiplication of the differential operator by a vector field:

The result of the action of this operator on a specific vector field F called field rotor F or, in short, just rotor F and represents a new vector field:

Rot field F(length and direction of vector rot F at each point in space) characterizes in a sense the rotational component of the field F respectively at each point.

Intuitive image

If v(x,y,z) is the field of gas velocity (or liquid flow), then rot v- a vector proportional to the angular velocity vector of a very small and light speck of dust (or ball) located in the flow (and carried away by the movement of gas or liquid; although the center of the ball can be fixed if desired, as long as it can rotate freely around it).

Specifically rot v = 2 ω , Where ω - this angular velocity.

For a simple illustration of this fact, see below.

This analogy can be formulated quite strictly (see below). The basic definition through circulation (given in the next paragraph) can be considered equivalent to that obtained in this way.

Mathematical definition

The curl of a vector field is a vector whose projection on each direction n is the limit of the relation of circulation of a vector field along a contour L, which is the edge of the flat area Δ S, perpendicular to this direction, to the size of this area, when the dimensions of the area tend to zero, and the area itself contracts to a point:

The direction of traversal of the contour is selected so that, when looking in the direction, the contour L walked clockwise.

In a three-dimensional Cartesian coordinate system, the rotor (as defined above) is calculated as follows (here F- denotes a certain vector field with Cartesian components, and - unit vectors of Cartesian coordinates):

For convenience, we can formally represent the rotor as a vector product of the nabla operator (on the left) and the vector field:

(The last equality formally represents the vector product as a determinant.)

Related definitions

A vector field whose curl is zero at any point is called irrotational and is potential. Since these conditions are necessary and sufficient for each other, both terms are practical synonyms. (However, this is true only for the case of fields defined on a simply connected domain).

For a little more detail about the mutual conditionality of potentiality and the irrotational nature of the field, see below (Basic properties).

On the contrary, a field whose curl is not equal to zero is usually called vortex , such a field cannot be potential.

Generalization

The most direct generalization of the rotor as applied to vector (and pseudovector) fields defined on spaces of arbitrary dimension (provided that the dimension of the space coincides with the dimension of the field vector) is as follows:

with indexes m And n from 1 to the dimension of space.

This can also be written as an external product:

In this case, the rotor is an antisymmetric tensor field of valence two.

In the case of dimension 3, convolution of this tensor with the Levi-Civita symbol gives the usual definition of a three-dimensional rotor given in the article above.

For a two-dimensional space, in addition, if desired, a similar formula with a pseudoscalar product can be used (such a rotor will be a pseudoscalar coinciding with the projection of the traditional vector product onto an axis orthogonal to the given two-dimensional space - if we consider the two-dimensional space to be embedded in some three-dimensional space, so that the traditional vector product has meaning).

The most important characteristics of a vector field are rotor and divergence. In this section we will consider the mathematical description of these characteristics of vector fields and methods for calculating them using differential operations. In this case, we will use only the Cartesian coordinate system. We will consider a more complete definition of divergence and rotor and their physical meaning in the next chapter. We will consider the calculation of these quantities in curvilinear coordinate systems later.

Let's consider a vector field defined in three-dimensional space.

Definition 1. The divergence of a vector field is a number that is defined by the expression

It is assumed that the corresponding partial derivatives exist at the point under consideration. The divergence of a vector field, just like the gradient, can be written using the nabla operator

Here divergence is represented as the scalar product of vectors and F. Let us note without proof that divergence describes the density of sources creating the field.

Example 1. Calculate the divergence of a vector field at a point.

Definition 2. The curl of a vector field is a vector that is defined by the expression

Note that in the presented sum, the indices in adjacent terms change according to the circular permutation rule, taking into account the rule.

The curl of a vector field can be written using the nabla operator

A rotor characterizes the tendency for a vector field to rotate or swirl, so it is sometimes called a vortex and is designated curlF.

Example 1. Calculate the curl of a vector field at a point.

Sometimes it becomes necessary to calculate the gradient of a vector field. In this case, the gradient from each component of the vector field is calculated. The result is a tensor of the second rank, which determines the gradient of the vector. This tensor can be described by the matrix

To describe such objects it is convenient to use tensor notation

believing. The use of tensor methods simplifies mathematical operations on such objects. A detailed presentation of the apparatus of tensor calculus is given in the course “Fundamentals of Tensor Analysis,” which is taught in parallel to the course “Additional Chapters of Higher Mathematics.”

Example 1. Calculate the gradient of a vector field.

Solution. For calculations we use tensor notation. We have

Here the Kronecker symbol is the identity matrix.

Example 2. Calculate the gradient of the scalar field and compare the expressions and.

Some properties of the nabla operator

Previously we introduced the vector differentiation operator

Using this operator, we wrote down the main differential operations in tensor fields:

The operator is a generalization of the differentiation operator and has the corresponding properties of the derivative:

1) the derivative of the sum is equal to the sum of the derivatives

2) the constant multiplier can be taken out of the operator sign

Translated into the language of vector functions, these properties have the form:

These formulas are derived in the same way as the corresponding formulas for the derivatives of a function of one variable.

Using the Hamilton operator allows us to simplify many operations related to differentiation in tensor fields. However, keep in mind that this operator is a vector operator and must be handled carefully. Let's look at some applications of this operator. In this case, the corresponding formulas are written both using the Hamilton operator and in conventional notation.

The concept of divergence as a local property of a vector field was introduced when considering the flow of a vector field on a closed surface. Similarly, one can introduce the corresponding characteristic when considering the circulation of a vector field.

Let's consider some point M and vector field a . Let us choose some direction characterized by the unit vector n and a plane perpendicular to the vector n and passing through the point M. Let's surround the point M contour L, lying in a given plane. Let us calculate the circulation of the vector field along this contour and take the ratio of this circulation to the area S, limited by contour L:

Let us now find the limit of this ratio at S®0, provided that the contour L shrinks to a point M without leaving the plane. This limit is called rotor vector field a at point M:

. (7.6)

Note 3. The rotor is a characteristic of the “rotational component” of the vector field, so it is denoted as rot. However, sometimes instead of the word rotor the term " vortex" and is designated by the symbol curl.

Let us now derive the formula for the rotor in the Cartesian coordinate system. Let n coincides with the direction of the axis Oz, and the contour L is a rectangle with sides D x and D y, while the circuit is traversed counterclockwise (see Fig. 7.3). Then we get

For the first term we get

(segments D.A. And B.C. can be ignored, since here x=const And dx=0). Further

Similarly we obtain for the second term

As a result, we find

Similarly, we calculate projections on other coordinate axes:

, .

In vector form this can be done as follows:

This formula can be written more compactly in symbolic form:

. (7.8)

Formula (7.7) is obtained from (7.8) by expanding the determinant along the first row.

Example 7.4. Calculate the curl of a vector field a =x 2 y 3 i +j +z k at point M(1;1;1).

Solution. Let's write it down

Thus,

Example 7.5. Find the rotor of the velocity field of the rotating body v =–w y i +w x j .

Solution. Because the v x=–w y, v y=w x, v z=0, then

So, the rotor of velocities of a rigid body at any point is equal to twice the angular velocity. The found mechanical meaning of the rotor has a broader meaning. For example, a wheel with blades in a fluid flow will have a maximum rotation speed if the axis of rotation is directed along rot a , and the rotation speed will be equal to .

Field theory

Also known as vector analysis. And for some, vector analysis, known as field theory =) Finally, we got to this interesting topic! This section of higher mathematics cannot be called simple, however, in future articles I will try to achieve two goals:

a) so that everyone understands what the conversation is all about;

b) and so that “dummies” learn to solve, at a minimum, simple things - at least at the level of tasks that are offered to part-time students.

All material will be presented in a popular style, and if you need more rigorous and complete information, you can take, for example, the 3rd volume of Fichtenholtz or look at Wiki.

And let’s immediately decipher the title. With the theory, I think everything is clear - in the best traditions of the site, we will analyze its basics and focus on practice. Well, what do you associate the word “field” with?

Grass field, football field…. More? Field of activity, field of experiments. Greetings humanists! ...From a school course? Electric field, magnetic, electromagnetic..., okay. The gravitational field of the Earth in which we find ourselves. Great! So, who said that about the field? valid And complex numbers? ...some monsters have gathered here! =) Thankfully algebra already passed.

In the next lessons we will get acquainted with a specific concept fields, specific examples from life, and also learn how to solve thematic problems of vector analysis. Field theory is best studied, as you correctly guess, in a field - in nature, where there is a forest, a river, a lake, a village house, and I invite everyone to immerse themselves, if not in the warm summer reality, then in pleasant memories:

Fields in the sense considered today are scalar And vector, and we will start with their “building blocks”.

Firstly, scalar. Quite often this term is mistakenly identified with number. No, things are a little different: scalar is a quantity, each value of which can be expressed just one number. There are a lot of examples in physics: length, width, area, volume, density, temperature, etc. All these are scalar quantities. And, by the way, mass is also an example.

Secondly, vector. I touched on the algebraic definition of a vector in the lesson about linear transformations and one of his private incarnations It’s simply impossible not to know=) Typical vector is expressed two or more numbers(with your coordinates). And even for a one-dimensional vector only one number not enough– for the reason that the vector also has a direction. And the point of application if the vector not single. Vectors characterize physical force fields, speed and many other quantities.

Well, now you can start harvesting aluminum cucumbers:

Scalar field

If each some point areas of space a certain number is assigned (usually real), then they say that in this area it is given scalar field.

Consider, for example, a perpendicular emanating from the earth Ray. Stick a shovel in for clarity =) What scalar fields can I ask on this beam? The first thing that comes to mind is height field– when each point of the beam is assigned its height above ground level. Or, for example, atmospheric pressure field– here each point of the beam corresponds to a numerical value of atmospheric pressure at a given point.

Now let’s approach the lake and mentally draw a plane over its surface. If each point of the “water” fragment of the plane is associated with the depth of the lake, then, please, the scalar field is given. At these same points, you can consider other scalar quantities, for example, the temperature of the water surface.

The most important property of a scalar field is his invariance relative to the coordinate system. If we translate it into human language, then no matter from which side we look at the shovel / lake - a scalar field (height, depth, temperature, etc.) this will not change. Moreover, the scalar field, say, depth, can be set on another surface, for example, on a suitable hemisphere, or directly on the water surface itself. Why not? Is it not possible to assign a number to each point of the hemisphere located above the lake? I suggested flatness solely for the sake of convenience.

Let's add one more coordinate. Take a stone in your hand. Each point of this stone can be assigned to its physical density. And again - no matter in what coordinate system we consider it, no matter how we twist it in our hand - the scalar density field will remain unchanged. However, some people may dispute this fact =) Such is the philosopher's stone.

From a purely mathematical point of view (beyond physical or other private meaning) scalar fields are traditionally specified by our “ordinary” functions one , two , three and more variables. At the same time, in field theory, traditional attributes of these functions are widely used, such as domain, level lines and surfaces.

With three-dimensional space everything is similar:
– here, each permissible point in space is associated with a vector with a beginning at a given point. “Admissibility” is determined by the domains of definition of functions, and if each of them is defined for all “X”, “E”, “Z”, then the vector field will be specified in the entire space.

! Designations : vector fields are also denoted by the letter or, and their components by or, respectively.

From the above it has long been clear that, at least mathematically, scalar and vector fields can be defined throughout space. However, I was still careful with the corresponding physical examples, since such concepts as temperature, gravity(or others) after all somewhere may not exist at all. But this is no longer horror, but science fiction =) And not only science fiction. Because the wind, as a rule, does not blow inside the stones.

It should be noted that some vector fields (same velocity fields) change rapidly over time, and therefore many physical models consider an additional independent variable. By the way, the same applies to scalar fields - temperature, in fact, is also not “frozen” in time.

However, within the framework of mathematics, we will limit ourselves to the trinity, and when such fields “meet” we will imply some fixed moment in time or a time during which the field has not changed.

Vector lines

If scalar fields are described lines and level surfaces, then the “shape” of the vector field can be characterized vector lines. Probably many remember this school experience: a magnet is placed under a sheet of paper, and on top (let's see!) iron filings spill out, which just “line up” along the field lines.

I’ll try to formulate it more simply: each point of a vector line is the beginning field vector, which lies on the tangent at a given point:

Of course, line vectors in the general case have different lengths, so in the above figure, when moving from left to right, their length increases - here we can assume that we are approaching, for example, a magnet. In force physical fields, vector lines are called - power lines. Another, simpler example is the Earth's gravitational field: its field lines are rays with the beginning in the center of the planet, and the vectors gravity located directly on the rays themselves.

Vector lines of velocity fields are called current lines. Imagine a dust storm again - dust particles along with air molecules move along these lines. Similarly with a river: the trajectories along which molecules of liquid (and not only) move are, in the literal sense, streamlines. In general, many concepts of field theory come from hydrodynamics, which we will encounter more than once.

If a “flat” vector field is given by a nonzero function, then its field lines can be found from differential equation. The solution to this equation gives family vector lines on a plane. Sometimes in tasks it is necessary to draw several such lines, which usually does not cause difficulties - we chose several convenient values of “tse”, drew some hyperboles, and order.

With the spatial vector field the situation is more interesting. Its field lines are determined by the relations . Here we need to decide system of two differential equations and get two families spatial surfaces. The intersection lines of these families will be spatial vector lines. If all components (“pe”, “ku”, “er”) are non-zero, then there are several technical solutions. I won't consider all these methods. (because the article will grow to indecent sizes), but I will focus on a common particular case, when one of the components of the vector field is equal to zero. Let's list all the options at once:

if , then the system needs to be solved;
if , then the system;
and if , then .

And for some reason we haven’t had practice for a long time:

Example 1

Find the field lines of the vector field

Solution: in this problem, therefore we solve system:

The meaning is very simple. So, if a function specifies a scalar field of lake depth, then the corresponding vector function defines the set unfree vectors, each of which indicates a direction speedy rise bottom at one point or another and the speed of this rise.

If a function specifies a scalar temperature field of a certain region of space, then the corresponding vector field characterizes the direction and speed fastest warm-up space at every point in this area.

Let's look at a general mathematical problem:

Example 3

Given a scalar field and a point. Required:

1) compose the gradient function of the scalar field;

Which is equal to potential difference .

In other words, in the potential field only the starting and ending points of the route matter. And if these points coincide, then the total work of forces along a closed contour will be equal to zero:

Let's pick up a feather from the ground and deliver it to the starting point. In this case, the trajectory of our movement is again arbitrary; you can even drop the pen, pick it up again, etc.

Why is the final result zero?

Did the feather fall from point “a” to point “b”? It fell. The force of gravity did the work.

Did the pen hit point "a" back? Got it. This means that exactly the same work was done against gravity, and it doesn’t matter with what “adventures” and with what forces - even if the wind blew him back.

Note : In physics, the minus sign symbolizes the opposite direction.

Thus, the total work done by the forces is zero:

As I have already noted, the physical and lay concept of work are different. And this difference will help you understand well not a feather or even a brick, but, for example, a piano :)

Together, lift the piano and lower it down the stairs. Drag it down the street. As much as you want and wherever you want. And if no one called the fool, bring the instrument back. Have you worked? Certainly. Until the seventh sweat. But from the point of view of physics, no work has been done.

The phrase “potential difference” is tempting to talk more about the potential electrostatic field, but shocking your readers is somehow not at all humane =) Moreover, there are countless examples, because any gradient field is potential, of which there are a dime a dozen.

But it’s easy to say “a dime a dozen”: here we are given a vector field - how to determine whether it is potential or not?

Vector field rotor

Or him vortex component, which is also expressed by vectors.

Let's take the feather in our hands again and carefully send it floating down the river. For the purity of the experiment, we will assume that it is homogeneous and symmetrical relative to its center. The axle sticks up.

Let's consider vector field current speed, and a certain point on the water surface above which the center of the feather is located.

If in at this point the pen rotates counterclockwise, then we will match it with the outgoing unfree upward vector. At the same time, the faster the pen rotates, the longer this vector is, ... for some reason it seems so black to me in the bright rays of the sun... If the rotation occurs clockwise, then the vector “looks” down. If the pen does not rotate at all, then the vector is zero.

Meet - this is it rotor vector vector velocity field, it characterizes the direction of “swirling” of the liquid in at this point and angular speed of rotation of the pen (but not the direction or speed of the current itself!).

It is absolutely clear that all points of the river have a rotary vector (including those that are “under water”), thus, for vector field of current velocity we have defined a new vector field!

If a vector field is given by a function, then its rotor field is given by the following vector function:

Moreover, if the vectors rotor field rivers are large in magnitude and tend to change direction, this does not mean at all that we are talking about a winding and restless river (back to the example). This situation can also be observed in a straight channel - when, for example, in the middle the speed is higher, and near the banks it is lower. That is, the rotation of the pen is generated different flow rates V neighboring current lines.

On the other hand, if the rotor vectors are short, then it could be a “winding” mountain river! It is important that in adjacent current lines the speed of the current itself (fast or slow) differed slightly.

And finally, we answer the question posed above: at any point in the potential field its rotor is zero:

Or rather, the zero vector.

Potential field is also called irrotational field.

An “ideal” flow, of course, does not exist, but quite often one can observe that velocity field rivers are close to potential - various objects float calmly and do not spin, ...did you also imagine this picture? However, they can swim very quickly, and in a curve, and then slow down, then speed up - it is important that the speed of the current is in adjacent current lines was preserved constant.

And, of course, our mortal gravitational field. For the next experiment, any fairly heavy and homogeneous object is well suited, for example, a closed book, an unopened can of beer, or, by the way, a brick that has waited in the wings =) Hold its ends with your hands, lift it up and carefully release it into free fall. It won't spin. And if it does, then this is your “personal effort” or the brick you got was the wrong one. Don't be lazy and check this fact! Just don't throw anything out of the window, it's not a feather anymore

After which, with a clear conscience and increased tone, you can return to practical tasks:

Example 5

Show that a vector field is potential and find its potential

Solution: the condition directly states the potentiality of the field, and our task is to prove this fact. Let's find the rotor function or, as they more often say, the rotor of a given field:

For convenience, we write down the field components:

and let's start finding them partial derivatives– it’s convenient to “sort through” them in a “rotary” order, from left to right:
- And straightaway check that (to avoid doing extra work in case of a non-zero result). Let's move on: