You can also use the "nabla" operator for the operation:

It is taken into account here that vector product collinear operators is equal to zero. It is proposed to obtain the same result by direct differentiation.

From the obtained result one can get important consequence. Consider some closed curve L and stretch an arbitrary surface onto it S.

Using Stokes' theorem, we can write

Let us formulate the result obtained in the form of a theorem:

Theorem 1. Circulation vector field along any closed contour is equal to zero.

Corollary 1. Curvilinear integral on the gradient of the scalar function does not depend on the choice of integration path and is completely determined by the initial and end points integration lines.

Proof. Let's make a drawing.

Let's perform the simplest transformations

Hence

This means that integrand is full differential. Consequently, the value of the integral depends only on the choice of points A and B:

Let's calculate the operation. To do this, we use the well-known vector algebra formula for double cross product

Let's rewrite this formula in a form more convenient for us

The transformation is made so that in further formulas the “nabla” operator does not appear in the last position. In terms of the operator "nabla" we get

(What would happen if we used the usual formula for the double cross product?)

Using the Laplace operator notation, we can write

We have a system of three differential relations written for the vector components F.

We looked at the basic second order differential operations. In the future we will use them to solve various problems.

Green's formulas

Let's get a few more formulas general, which relate properties various functions and are widely used in applications. Let's write the Gauss-Ostrogradsky formula

Let and be two arbitrary scalar functions. Let's put

Then the Gauss-Ostrogradsky theorem takes the form

You can write down

Here the notation is introduced

for the derivative of a function in direction

After substituting these expressions into the modified Gauss-Ostrogradsky formula, we obtain

This formula is called Green's first formula.

Similarly, if we put

then Green's first formula takes the form

Subtracting corresponding formulas, we get

This formula is called Green's second formula.

Using Green's formulas, it is possible to obtain connections between the values ​​of the function at the internal points of the selected volume and at the boundaries.

Theorem 1. The value of the function in internal point region T, limited by the surface S, is determined by the formula

distance between points and. Proof. Consider a point and surround it with a small one spherical surface radius

1. Basic concepts of field theory

Field theory underlies many concepts modern physics, mechanics, mathematics. Its main concepts are gradient, flow, potential, rotor, divergence, circulation, etc. These concepts are also important in mastering the basic ideas mathematical analysis functions of many variables.

A field is a region G of space, at each point of which the value of a certain quantity is determined.

IN physical problems There are usually two types of quantities: scalars and vectors. In accordance with this, two types of fields are considered.

If each point M of this area is associated with a certain number U(M), they say that in

area is given (defined) a scalar field. Examples of scalar fields are the temperature field inside some heated body (at each point M of this body the corresponding temperature U (M) is specified), the field

illumination created by any light source. Let the system be fixed in space

coordinates of point M in this coordinate system. The values ​​of the functionU(x,y,z) coincide with the values ​​of the fieldU(M),

therefore, the same symbol is retained for it.

If each point M of this area is associated with a certain vector (M), they say that

a vector field is specified. One example of vector fields is the velocity field of a stationary fluid flow. It is defined as follows: let region G be filled with liquid flowing at each point with

some speed v, independent of time (but

different, generally speaking, in different points); By assigning vectorv (M) to each point M from G, we obtain a vector field called the velocity field.

If a(M) is some vector field in

space, then taking a fixed rectangular Cartesian coordinate system in this space, we can

represent a(M) as an ordered triple of scalar

functions: a (M) = (P (x,y,z),Q (x,y,z),R (x,y,z)). These

If the function U (M) (or a (M)) does not depend on

time, then the scalar (vector) field is called stationary; a field that changes over time is called non-stationary. Below we will consider only stationary fields.

2. Basic characteristics of scalar and vector fields

A vector whose coordinates are the values ​​of the partial derivatives of the function U (x,y,z) at the point

M (x ,y ,z ) is called the gradient of the function and denotes

It is known that the gradient at point M sets the direction of the fastest increase in the function U (x,y,z). They say that the scalar field U generates

vector gradient field U.

Gradient line scalar field U(M) is called

any curve whose tangent at each point is directed along gradU at that point.

Thus, the field gradient lines are those lines along which the field changes most rapidly.

To formulate another property of a gradient, let us recall the definition of a level surface.

Level surface functions (fields)U =U (x,y,z)

is the surface on which the function (field) preserves constant value. The level surface equation has the form U (x,y,z) =C.

Thus, at each point in the field, the gradient is directed along the normal to the level surface passing through this point.

Flow Π of the vector fielda = (P ,Q ,R ) through

the surface σ is called surface integral

∫∫ (P cosα + Q cosβ + R cosγ )dS

or, in short, ∫∫ a n dS, where through n = (cosα, cosβ, cosγ)

designated unit vector normal to the surface σ, defining its side.

are continuous, then

3. Stokes and Ostrogradsky-Gauss formulas

Let us denote by L a certain closed contour, and σ the surface spanned by this contour.

It is assumed that the choice of direction on the contour is consistent with the choice of the side of the surface (when traversing the contour in the selected direction, the selected side is on the left).

The Stokes formula says that the circulation of a vector field along a certain contour is equal to the flux of the vector field rotor through a surface stretched over this contour.

Let now Ω be some closed limited area, aσ is the boundary of this area. Then it is fair

σ Ω

Recall that the surface integral on the left side of formula (5) is taken according to outside surface σ.

The Ostrogradsky-Gauss formula means that triple integral over the area from the divergence of the vector field equal to the flow of this field through the surface bounding this area.

4. Hamilton operator. Some types of scalar and vector fields

The English mathematician and mechanic Hamilton introduced the vector differential operator

called the nabla operator.

It should be immediately noted that the analogy between a symbolic vector and “real” vectors is not

complete. Namely, formulas containing a symbolic vector are similar to ordinary vector algebra formulas if they do not contain products variables(scalar and vector), that is, until you have to apply the differentiations included in the operations to the product of variable quantities.

Using nabla vector, scalar field gradient

The expediency of introducing a symbolic vector lies in the fact that with its help it is convenient to obtain and write various formulas vector analysis.

Let us demonstrate this with examples.

Problem 1. Prove that the rotor of the gradient of the scalar field U (M) is equal to 0, that is, rot(gradU) = 0.

Let us first prove this equality without using the Hamilton operator. Thus,

∂ ∂

= ∂ x∂ y∂ U∂ U

∂x∂y

since, by Schwarz's theorem, continuous mixed derivatives are equal.

Now, using the form of writing the gradient (7) and the rotor (9) through, we have rot(gradU ) =× U .

Since the vector U (the product of a vector and the scalar U) is collinear to the vector, then their vector

the product is 0.

Task 2. Write the divergence of the scalar field gradient div(gradU ) using.

Forming a divergence from gradU, we get

= ∂ 2 U + ∂ 2 U + ∂ 2 U . ∂ x 2∂ y 2∂ z 2

Laplace and is denoted by the symbol:

= ∂ 2+ ∂ 2+ ∂ 2. ∂ x 2∂ y 2∂ z 2

Since the scalar square of a vector equal to square its modulus, then = 2. Thus, div(gradU ) =2 U .

The vector field a (M) is called potential,

if it can be represented as the gradient of some scalar field U(M):

a = gradU .

The scalar field U itself is called the vector field potentiala.

In order for the vector field a(M) to be

The necessity of fulfilling equality (10) has been proven (see Problem 1 discussed above).

The vector field potential can be found using the formula

where (x 0 ,y 0 ,z 0 ) - arbitrary point areas G .

Vector field a(M), whose divergence

identically equal to zero, is called solenoidal (tubular).

In order to formulate one of the most important properties solenoidal field, we introduce the concepts of vector line and vector tube.

A line L lying in G is called a vector

line if at each point of this line the direction of the tangent to it coincides with the direction of the vector field at this point.

It is known that a vector line is an integral curve of a system of differential equations

In particular, if a vector field is a velocity field of a stationary fluid flow, then its vector lines are the trajectories of fluid particles.

A vector tube is a closed set Φ of points in a region G, in which a vector field a (M) is specified, such that everywhere on its boundary surface the normal vector n is orthogonal to (M).

A vector tube consists of vector field lines a(M). A vector line is entirely contained in Φ if

one point of the line is contained in Φ.

The intensity of the tube Φ in a section is the field flux (M) through this section.

If the field is solenoidal, then the law of conservation of vector tube intensity is satisfied.

For the velocity field v(M) of an incompressible fluid in the absence of sinks and sources (that is, under the condition divv(M) = 0), the law of conservation of vector intensity

tubes can be formulated in this way: the amount of liquid flowing per unit time through a cross section of a vector tube is the same for all its sections.

Below are some typical tasks with solutions.

Task 3. Find scalar field level surfaces

U (M) = x2 + y2 − z.

level surfaces are a family of elliptical paraboloids whose axis of symmetry is the Oz axis.

Task 4.

In the scalar field U (M ) = xy 2 + z 2 find

gradient at point M 0 (2,1,− 1) .

Let's find the values

partial derivatives

U (M) at point M 0:

∂U

|M 0 =y 2 |M 0 = 12 = 1,

∂U

|M 0 = 2xy |M 0 = 2 2 1 = 4,

∂x

∂y

∂U

2 (− 1) =− 2.

∂z

Hence,

gradU (M 0 ) =s i + 4s j − 2k s .

Calculate the divergence of a vector field

a(M) = 2 xy2 i− yz j+ 3 z2 k

at point M 0 (1,− 2,1) .

P = 2xy 2 ,Q =− yz ,R = 3z 2 . Let's find the value

corresponding partial derivatives at the point M 0:

∂P|

2 y 2 |

2 (− 2)2 = 8,

∂Q

= − z |

= − 1,

∂x

∂y

Rotor (mathematics) Rotor, or vortex is a 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 entrained 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 three dimensions Cartesian system coordinates 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 rotor equal to 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 usual definition 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 the 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 paragraph we will look at mathematical description these characteristics of vector fields and methods for calculating them using differential operations. In this case, we will use only the Cartesian coordinate system. More full definition divergence and rotor and their physical meaning We'll look at it 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 the divergence is represented as dot product 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. Using tensor methods simplifies mathematical operations over 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 have written down the basic 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) constant factor can be taken out as an 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. Did you like the article? Share with your friends! Share on Facebook Read also Planets of the solar system: their order and history of names Choosing a Career (Teacher) Chapter ten Nikolai Yavorsky - founder of the ballet school in Cuba A school where children love to learn and teachers love to teach Manifesto for the liberation of the peasants Both peasants and landowners were dissatisfied Ural Cossacks - anti-Bolshevik struggle and exodus to Persia Where the Cossacks run everything Yusupov. Incredible stories. No, I don’t dare compare myself with a genius Prince Boris Nikolaevich Yusupov Farewell in German - German online - Start Deutsch Greetings in a letter in German Graph theory Application of graph theory in chemistry Top