Ostrogradsky-Green formula
This formula establishes a connection between the curvilinear integral over a closed contour C and the double integral over the area limited by this contour.
Definition 1. A region D is called a simple region if it can be divided into a finite number of regions of the first type and, independently of this, into a finite number of regions of the second type.
Theorem 1. Let functions P(x,y) and Q(x,y) be defined in a simple domain and be continuous along with their partial derivatives and
Then the formula holds
where C is the closed contour of area D.
This is the Ostrogradsky-Green formula.
Conditions for the independence of a curvilinear integral from the path of integration
Definition 1. A closed squarable region D is said to be simply connected if any closed curve l D can be continuously deformed into a point so that all points of this curve would belong to the region D (region without “holes” - D 1), if such deformation is impossible, then the region is called multiply connected (with “holes” - D 2).
Definition 2. If the value of a curve integral along a curve AB does not depend on the type of curve connecting points A and B, then this curve integral is said to be independent of the path of integration:
Theorem 1. Let the continuous functions P(x,y) and Q(x,y) be defined in a closed simply connected domain D, together with their partial derivatives. Then the following 4 conditions are equivalent:
1) curvilinear integral over a closed loop
where C is any closed loop in D;
2) the curvilinear integral over a closed loop does not depend on the path of integration in region D, i.e.
3) the differential form P(x,y)dx + Q(x,y)dy is the total differential of some function F in the domain D, i.e., there is a function F such that (x,y) D the equality holds
dF(x,y) = P(x,y)dx + Q(x,y)dy; (3)
4) for all points (x,y) D the following condition will be satisfied:
Let's prove it using the diagram.
Let's prove that from.
Let 1 be given), i.e. = 0 by property 2 §1, which = 0 (by property 1 §1) .
Let's prove that from.
It is given that cr.int. does not depend on the path of integration, but only on the choice of the beginning and end of the path
Consider the function
Let us show that the differential form P(x,y)dx + Q(x,y)dy is the complete differential of the function F(x,y), i.e. , What
Let's set the private growth
x F (x,y)= F(x + x, y) -F (x,y)= = == =
(by property 3 of § 1, BB* Oy) = = P (c,y)x (by the mean value theorem, c -const), where x (due to the continuity of the function P). We obtained formula (5). Formula (6) is obtained similarly. Let's prove that from. The formula is given dF(x,y) = P(x,y)dx + Q(x,y)dy. Obviously = P(x,y). Then According to the conditions of the theorem, the right-hand sides of equalities (7) and (8) are continuous functions, then, by the theorem on the equality of mixed derivatives, the left-hand sides will also be equal, i.e., that Let us prove that out of 41. Let us choose any closed contour from the region D that bounds the region D 1 . The functions P and Q satisfy the Ostrogradsky-Green conditions: By virtue of equality (4), on the left side of (9) the integral is equal to 0, which means that the right side of the equality is also equal to Remark 1. Theorem 1 can be formulated in the form of three independent theorems Theorem 1*. In order for a simply connected squarable domain D to have a curved int. did not depend on the path of integration so that condition (.1) is satisfied, i.e. Theorem 2*. In order for a simply connected squarable domain D to have a curved int. did not depend on the path of integration so that condition (3) is satisfied: the differential form P(x,y)dx + Q(x,y)dy is the total differential of some function F in the domain D. Theorem 3*. In order for a simply connected squarable domain D to have a curved int. did not depend on the path of integration so that condition (4) is satisfied: Remark 2. In Theorem 2*, the domain D can also be multiply connected. (Ostrogradsky Mikhail Vasilievich (1861–1862) - Russian mathematician, Academician Petersburg A.N.) (George Green (1793 – 1841) – English mathematician) Sometimes this formula is called Green's formula, however, J. Green proposed in 1828 only a special case of the formula. The Ostrogradsky–Green formula establishes a connection between the curvilinear integral and the double integral, i.e. gives an expression for the integral over a closed contour in terms of the double integral over the region limited by this contour. If the closed contour has the form shown in the figure, then the curve integral along the contour L can be written as: If sections AB and CD of the contour are taken as arbitrary curves, then, after carrying out similar transformations, we obtain a formula for a contour of arbitrary shape: This formula is called the Ostrogradsky–Green formula. The Ostrogradsky–Green formula is also valid in the case of a multiply connected region, i.e. area within which there are excluded areas. In this case, the right side of the formula will be the sum of the integrals along the outer contour of the region and the integrals along the contours of all excluded sections, and each of these contours is integrated in such a direction that the region D remains on the left side of the bypass line at all times. Example. Let us solve the example discussed above using the Ostrogradsky–Green formula. The Ostrogradsky–Green formula allows you to significantly simplify the calculation of the curvilinear integral. The line integral does not depend on the shape of the path if it has the same value along all paths connecting the starting and ending points. The condition for the independence of a curvilinear integral from the shape of the path is equivalent to the equality to zero of this integral along any closed contour containing the starting and ending points. This condition will be satisfied if the integrand is a complete differential of some function, i.e. the condition of totality is satisfied. These formulas connect the integral over a figure with some integral over the boundary of a given figure. Let the functions be continuous in the domain DÌ Oxy and on its border G; region D– connected; G– piecewise smooth curve. Then true Green's formula: here on the left is a curvilinear integral of the first kind, on the right is a double integral; circuit G goes counterclockwise. Let T– piecewise smooth bounded two-sided surface with piecewise smooth boundary G. If the functions P(x,y,z), Q(x,y,z), R(x,y,z) and their first order partial derivatives are continuous at points of the surface T and borders G, then it occurs Stokes formula: on the left is a curvilinear integral of the second kind; on the right – surface integral of the second kind, taken along that side of the surface T, which remains to the left when traversing the curve G. If a connected region WÌ Oxyz bounded by a piecewise smooth, closed surface T, and the functions P(x,y,z), Q(x,y,z), R(x,y,z) and their first order partial derivatives are continuous at points from W And T, then it occurs Ostrogradsky-Gauss formula: on the left – surface integral of the second kind over the outer side of the surface T; on the right – triple integral over the area W. Example 1. Calculate the work done by the force Solution. Work is equal to Example 2. Calculate integral Solution. Using the Stokes formula (2.23), we reduce the original integral to the surface integral over a circle T: So, given that , we have: The last integral is a double integral over a circle DÌ Oxy, on which the circle was projected T; D: . Let's move on to polar coordinates: x=r cosj, y=r sinj, jÎ, r O. Eventually: Example 3. Find stream P T pyramids W: Solution. The flow is Example 4. Find stream P vector field through the complete surface T pyramids W: ; (Fig. 2.20), in the direction of the outer normal to the surface. Solution. Let us apply the Ostrogradsky-Gauss formula (2.24), where V– volume of the pyramid. Let's compare with the solution of direct calculation of the flow ( – faces of the pyramid). Communication between the two Int. In area D and curvilinear. Int. For region L, the Ostrogradsky-Green formula is established. Let the area D limit be specified on the OXY plane. A curve intersecting with straight parallel cords. The axes are at no more than 2 points, i.e. region D is correct. T1.If f. P(x,y), Q(x,y) is continuous along with its derivatives, Region D then has fair forms. L is the boundary of the region D and integration along the curve L is performed in the positive direction. Dovo. T2.If = (2), then the subintegrator. The expression P*dx+Q*dy yavl. Full differential Functions U=U(x,y). P*dx+Q*dy =U(x.y) Satisfies condition (2) can be found using f. Note 1 In order not to confuse the variable integral. X with the upper foreword is its designation. Another letter. Deputy 2 the point (0.0) is usually taken as the starting point (x0,Y0) Condition for the independence of a curvilinear int. 2nd kind from path integra. Let t. A (X1, Y1), B (X2, Y2),. Let the prod. points of area D. Points A and B can be connected by different lines. For each of them cr. Int. will have its own value if the value along all curves is the same, then the integral does not depend on the type of path int., in this case it is enough to note the initial. Point A (X1, Y1) and end point B (X2, Y2). T. In order to cr. Int. Does not depend on the path int. Area D in the cat. F. P(X,Y), Q(X,Y) are continuous along with their derivatives and it is necessary that at each point of the domain = Doc. Kr. Int. 2nd kind does not depend on the integration path Deputy = from here we get that Pov. Int. 1st kind.His St. and calc. Let at the points S S PL. S space oxyz def. Continuous f. f(x,y.z) . Let's break up the pov. S into n parts Si, PL. EACH PART has delta Si, and diameter Di i=1..m in each part Si, choose an arbitrary point Mi from (xi, yi, zi) and compile the sum . The sum is called integral for f. f(x,y.z) over the surface S if at integral. The amount has a limit, it is called. By integral of the 1st kind from f. f(x,y.z) over the surface S and is denoted = Properties of the surface Int. 2) Calculation pov int of the 1st kind reduce to calculating the 2nd int over the region D, which is the projection of the surface S onto the oxy plane, if the surface s is given Ur z=z(x,y) then the screw is equal to . If S is given in the form y=y(x, z), then... Pov int 2nd kind Let a two-sided surface be given; after going around such a surface without crossing its boundaries, the direction of the normal to it does not change. One-sided pov: is a Mobius strip. Let at a point of the considered two-sided surface S in the space oxyz be defined by f. F(x,y,z). We divide the selected side of the surface into parts Si i=1..m and project them onto the cord of the plane. In this case, we take the pl pov with the “+” sign if the top side of the pov is selected (if the normal forms an acute angle with oz, select with the “–” sign if the bottom side of the pov is selected (OBTITUDE ANGLE)). Let's compose an int sum Where - pl pov Si -parts with if it exists and does not depend on the method of dividing the surface into parts and on the choice of points in them, called int of the 2nd kind from f. f(x,y,z) on surface s and is denoted: Limited to this surface: that is integral of the divergence of a vector field extended over a certain volume T, is equal to the vector flux through the surface S, limiting this volume. The formula is used to convert a volume integral into an integral over a closed surface. In Ostrogradsky’s work, the formula is written as follows: where ω and s- volume and surface differentials, respectively. In modern notation ω = dΩ - volume element, s = dS
- surface element. - functions that are continuous together with their first-order partial derivatives in a closed region of space bounded by a closed smooth surface. A generalization of Ostrogradsky's formula is the Stokes formula for manifolds with boundary. The general method of converting a triple integral to a surface integral was first shown by Carl Friedrich Gauss (, gg.) using the example of electrodynamics problems. In 1826, M. V. Ostrogradsky derived the formula in general form, presenting it in the form of a theorem (published in 1831). A multidimensional generalization of the formula by M. V. Ostrogradsky was published in 1834. Using this formula, Ostrogradsky found the expression for the derivative with respect to the parameter of n-fold integral with variable limits and obtained a formula for variation n-multiple integral. Abroad the formula is called Gauss formula or “the formula (theorem) of Gauss-Ostrogradsky.” Wikimedia Foundation. 2010. Ostrogradsky formula Gauss-Ostrogradsky formula- Ostrogradsky Gauss theorem is a statement of integral calculus of functions of many variables, establishing a connection between n multiple integral over a domain and (n − 1) multiple integral over its boundary. Let V = (v1,v2,...,vn) be a vector field... ... Wikipedia Stokes formula- Stokes' theorem is one of the main theorems of differential geometry and mathematical analysis on the integration of differential forms, which generalizes several theorems of analysis. Named after J. G. Stokes. Contents 1 General formulation 2… … Wikipedia Green's formula- Green's theorem establishes a connection between a curvilinear integral over a closed contour C and a double integral over a region D bounded by this contour. In fact, this theorem is a special case of the more general Stokes theorem. The theorem is named in ... Wikipedia Liouville-Ostrogradsky formula- Liouville Ostrogradsky formula is a formula connecting the Wronski determinant (Wronskian) for solutions of a differential equation and the coefficients in this equation. Let there be a differential equation of the form y(n) + P1(x)y(n − 1) + P2(x)y(n − 2) ... Wikipedia Liouville formula- Ostrogradsky formula connecting the Wronski determinant (Wronskian) for solutions of a differential equation and the coefficients in this equation. Let there be a differential equation of the form then where is the Wronski determinant For a linear... ... Wikipedia OSTROGRAD FORMULA- a formula for the integral calculus of functions of many variables, establishing a connection between an n multiple integral over a domain and (n 1) a multiple integral at its boundary. Let the functions Xi=Xi(x1,x2,..., xn).together with their partial derivatives, i=1, 2... Mathematical Encyclopedia OSTROGRAD FORMULA- connects the triple integral (see Multiple integral) over a certain volume with the surface integral over the surface bounding this volume. Proposed by M. V. Ostrogradsky (1828 31) ... Big Encyclopedic Dictionary Ostrogradsky formula- a formula that transforms the integral taken over the volume Q, limited by the surface S, into the integral taken over this surface: here X, Y, Z are functions of the point (x, y, z) belonging to the three-dimensional region Ω. O. f. found... Great Soviet Encyclopedia
(2.23)
(2.24)
when traversing the point of its application of the circle G: , starting from the axis Ox, clockwise (Fig. 2.18).
. Let’s apply Green’s formula (2.22), placing the “-” sign on the right before the integral (since the circuit is traversed clockwise) and taking into account that P(x,y)=x-y, Q(x,y)=x+y. We have:
,
Where S D- area of a circle D: , equal to . As a result: – the required work of force.
, If G there is a circle in the plane z=2, going around counterclockwise.
T: 
.
(Fig. 2.19) in the direction of the outer normal to the surface.
. Applying the Ostrogradsky-Gauss formula (2.24), we reduce the problem to calculating the triple integral over the figure W-pyramid:
,
since the projection of faces onto the plane Oxy has zero area (Fig. 2.21),
(f.Ostr.-Gr.)
3) S=s1+s2, Then 4) f1<=f2 , т о 5) 6) 7) Ф. f непрерывна на поверхности S , то на этой поверхности сущ. Точка M(x0,y0,z0) S, такая, что 
.
By definition, the integral will be = the limit of the integral sum. Similarly, define int by pov s
, then the general form of a surface int of the 2nd kind is int where P, Q, R are continuous functions defined at the points of a two-sided surface s. If S is a closed surface, then by int along the outer side is also denoted along the inner side. ds. Where ds is the element of area pov S, and cos, cos cos for example cos is normalized to n. Selected side in turn.
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