Advanced Mathematics: Tenth chapter curve integral and curved area divide (3) Gauss consensus, flux, divergence, stokes consensus, circulation volume, curl _ higher mathematics

Flux and divergence of §10.6 Gauss formula

One, Gauss formula

The green formula expresses the relationship between the double integral on the plane closed area and the curve integral on its boundary curve, while the Gauss formula expresses the relationship between the triple integral on the spatial closed region and the curvature area on its boundary surface, which can be stated as follows:

The "theorem" set space closed area is formed by the piecewise smooth closed surface, function, and in the first order of continuous partial derivative, there are

(1)

or ()

Here is the outer side of the entire boundary surface, the direction cosine of the normal vector at the point, and the formula (1) or () is called the Gauss formula.

Syndrome: The relationship between the two-class curvature area, the formula (1) and the right end of () is equal, so as long as the formula (1) can be proved.

The projection area of the closed area on the plane is assumed to be the intersection point between the straight line and the boundary surface of the inner and parallel axis is exactly two. In this way, can be set by, and three parts, wherein and respectively by the equation and given, here, remove the side, take the upper side, is based on the boundary curve and bus parallel to the axis of the cylinder part of the outer side.

According to the calculation method of Sanchong integral, there

(2)

Because the projection of any surface on the plane is zero, it is directly based on the definition of the curvature area of the coordinate

Add the above three-style,

(3)

Comparison (2), (3) two-type,

If you cross a straight line that is parallel to an axis and a line that is parallel to an axis and the intersection point of the boundary surface is exactly two points, you can similarly

The above three-type two ends are added separately, namely the Gauss formula (1).

In the above proof, we make the limit of the closed region, that is, through the internal and parallel to the axis of the line and the boundary surface intersection is exactly two points. If this condition is not met, several auxiliary surfaces can be introduced into a finite closed region, so that each closed region satisfies such conditions and notes that the absolute value of the two curved area points along the opposite side of the auxiliary surface is equal and the symbol is reversed, so the formula (1) is still correct for such a closed region.

"Example 1" uses the Gauss formula to calculate the curved area Division

The outer side of the entire boundary curve of the cylindrical surface and the surrounding space area.

Solution: Here,,

，，，

Using the Gauss formula, the given curved surface area is converted into triple integral, and then the triple integral is computed by cylindrical coordinates:

"Example 2" uses the Gauss formula to calculate the curved area Division

Where the cone is between the plane and the part of the lower side, and is at the point of the normal vector of the direction of cosine.

Solution: Surface is not a closed surface, can not directly use the Gauss formula, supplementing the surface

Together with the formation of a closed surface, remember that they are surrounded by the space closed area, using the Gauss formula, there are

Among them, note that the

is to

and

So

"Example 3" sets the function and has the first order and the second order partial derivative in the closed region, the test proof

This is the entire boundary surface of the closed interval, the directional derivative of the function along the outer normal direction, which is called Green's first formula.

Syndrome: in the Gauss formula

, the order,,, and substituting the left and right sides of the upper type, and then get

The above two-type combination is the green first formula to be proved.

* Ii. conditions for dividing the curved area along any closed surface into 0

Set is the starting point, the end of the smooth curve, for the curve integral

If the integrand is satisfied, the above curve integral has nothing to do with, but only coordinates with.

Naturally, we will think of such a problem, under what conditions, the curvature of the area

is independent of the surface and is related only to the boundary curve. This problem is equal to the conditions under which the curvature area along any closed surface is divided into 0. This problem can be solved by Gauss formula.

First, we introduce the concept of space two-dimensional single connected and one dimensional single connected region.

For a space region, if the area enclosed by any closed surface belongs to the space, then it is called a two-dimensional single connected region; If any of the closed curves can always be an entirely curved surface, it is called a space one-dimensional single connected region.

For the condition of dividing the curved area along any closed surface into 0, we have the following conclusions:

"Theorem" is a space two-dimensional single connected region, the function, and the other has a first-order continuous partial derivative, the curved area

The sufficient and necessary condition of the boundary curve (or the curvature area of any closed surface along the inner part of 0) that is independent of the taken surface and is only dependent on is the equation

(4)

In the establishment of the inner constancy.

Syndrome: If the equation (4) is set up, then the Gaussian formula (1) can immediately see that the curvature area of any curved surface is divided into 0, so the condition (4) is sufficient.

Conversely, the curvature of any closed surface within the set is divided into 0, if the equation (4) is not constant, that is, at least one point makes

Following the method used in the second eye of section 10.3, it is possible to conclude that there is a closed surface in which the curvature of the closed surface is not equal to zero, which is contradictory to the hypothesis. The conditions (4) are therefore necessary.

Iii. Flux and divergence

Here's how to explain the Gauss formula

(1)

of physical significance.

A velocity field with a steady flow of incompressible liquids (assuming a density of 1) is provided by

Given, which, and assumed to have a first-order continuous partial derivative, is a moving surface in the velocity field, but also at the point of the unit normal vector, as discussed in section 10.5, the unit time fluid flow through the specified side of the total mass of fluid can be expressed by the curvature of the surface:

If it is the outer side of the boundary surface of the closed region in the Gauss formula (1), then the right end of the formula (1) can be interpreted as the total mass of the fluid leaving the closed region in the unit time.

Since we assume that the fluid is incompressible and the fluid is stable, when the fluid leaves, the interior must have the "source" of the fluid to produce the same number of fluids to supplement it. Therefore, the left end of the Gauss formula can be interpreted as the total mass of the fluid produced in the unit time by the source of the distribution.

For simplicity, the Gauss formula (1) is rewritten as

In order to remove the upper-type ends by the volume of the Closed area,

The upper-type left represents the average of the fluid mass produced by the source within the unit time unit volume. Applying the mean value theorem of integrals to the upper left end,

Here is a point within.

To indent a point, to take the upper-style limit, to

The upper-type left called the divergence at the point, which is recorded as

Here it can be seen as the fluid quality produced in the source Strength-unit time unit volume of a steady flow incompressible fluid at the point. If negative, the fluid at the point disappears.

In general, a vector field is set by

Given, among them, with the first-order continuous deflection, is a field of a directed surface, is the point of the unit normal vector, then called the vector field through the curvature of the specified side of the flux (or flow), and called the divergence of the vector field, that is,

The Gauss formula can now be written

Which is the boundary surface of the space closed area, and

is the projection of the vector on the lateral normal vector of the surface.

Loop flow and curl of §10.7 Stokes formula

One, Stokes formula

The Stokes formula is the generalization of Green's formula. The green formula expresses the relationship between the double integral on the plane closed area and the curve integral on its boundary curve, while the Stokes formula links the curved area on the surface with the curve integral along the boundary curve.

We first introduce the forward regulation of the boundary curve of the curved surface, and then state and prove the Stokes formula.

"Theorem" is a piecewise smooth space with a direction-closed curve, which is to assume that the forward and side of a piecewise smooth and curved surface of the boundary conforms to the right hand rule, a function, a space region containing a surface has a first order continuous partial derivative,

(1)

The formula (1) is called the Stokes formula.

Syndrome: First of all, it is assumed that there is no more than a straight line parallel to the axis, and the upper side of the curved surface is projected on the plane with a planar direction curve, and the closed region is formed.

We managed to divide the area of curvature

To the double integral on the closed area, and then through the green formula, it is connected to the curve integral.

Based on the relationship between the area and the curvature area of the coordinates, there

(2)

It is known from section 8.6 that the direction of the normal vector of the directed surface Yu Yingwei

，，

Therefore, the generation of it into (2) is

That

(3)

When the curved area of the upper right end is differentiated into a double integral, the middle one should be substituted, because by the differential method of the compound function, there

So, (3) The formula can be written

According to Green's formula, the double integral of the upper right end can be reduced to the curve integral of the boundary along the closed region.

So

Because the value of a function at the point on the curve is the same as the value of the function at the corresponding point on the curve, and the corresponding small arc on the two curves on the axis of the projection is the same, according to the definition of curve integral, the upper right end of the curve integral equal to the curve integral, therefore, we are proof

(4)

If the side is removed, the opposite direction is changed accordingly, and the end (4) type is simultaneously changing the symbol, so the (4) formula is still valid.

Second, if the surface intersects more than one line parallel to the axis, it can be used as an auxiliary curve to divide the surface into several parts, and then apply the formula (4) and add it. This is true for this type of curved surface formula (4), because the two curve integrals along the auxiliary curve are added in the same way.

The same can be verified

Adding them to the formula (4) is the formula (1).

To facilitate memory, use the determinant notation to write the Stokes formula (1)

The determinant is expanded by the first line, and the "product" is understood as the "product", which is "equal to" and so on.

This is exactly the integrand expression at the left end of the formula (1).

Another form of Stokes formula can be obtained by using the relationship between two types of curved area:

Which is the unit normal vector of the direction surface.

If it is a plane closed area on the plane, the Stokes formula becomes Green's formula. Therefore, the green formula is a special case of the Stokes formula.

"Example 1" using Stokes formula to calculate curve integral