The surface area of §10.4

I. Introduction of the Concept

1. Introduction Example

We know that if you have a thin sheet with mass density on the surface, the mass of the plane slice can be represented by the following double integral

When it is a space surface with mass density, it also has mass, so how does it define and calculate its mass? Obviously, the solution to this problem is still available in the element method we have used repeatedly.

Using any set of curves to divide a surface into a block "small" surface (representing both the block "small" surface and its area), if the surface density function is in continuous condition, the mass of the block "small" surface is approximately

Which is on any point.

Thus, the total mass of the surface is approximately

Thus

This represents the largest of the small surface diameter of a block.

Apart from the concrete meaning of this practical problem, the mathematical features that it contains are the following concepts of area curvature.

2, the area of the surface area of the definition

The space surface is smooth, the function is bounded, and any set of curves will be divided into small blocks (also representing the area of the small piece of surface), arbitrarily set a point on the small block, and type

By remembering the largest of each small surface diameter, if at that time the limit of the above and the formula exists, the limit value is called the curvature area of the function on the surface, and is recorded as

, i.e.

Where: called integrand, called Integral surface, is called surface area element.

According to the above definition, the surface density is a continuous function of the smooth surface of the mass, can be expressed as a function of the area on the surface of the curved surface

The surface area of the area is physically represented by the total mass of the mass distribution surfaces, which is the same as the physical meaning represented by the double integral, so it has some properties similar to that of the double integral.

The properties of the surface area of the area

1. Existence theorem

If the surface is smooth and the function is contiguous on the surface, the

Exist.

2, the surface can be additive

It can be divided into two smooth surfaces and (notes), while

And

exist, they exist, and there are

3,

4, if on the surface, there, then

Calculation method of surface area division of Area

The surface is given by the equation, the projection area on the plane is, the function has the first order continuous partial derivative, and the integrand is continuous.

According to the area of the curved area of the definition has

As shown in the figure, the section of the surface (its area is also recorded) on the plane projection area for (its area is also recorded), it can be expressed as

Point belongs to the surface, so

Thus, integrals and expressions can be expressed as

Because functions and functions are contiguous in a closed area, at that time, the limits of the upper-right end and

The limit is equal, equal to the double integral

So

This is the calculation formula of the double integral of the curved area differentiation of the area.

A memory method of formula

In the second, the integrand is defined, and its independent variable value should satisfy the equation.

In a similar way, we can derive two other formulas for calculating the surface area of the area.

"Example 1", for the cone in between the part.

The equation of "solution One" is

Its projection area on the plane is

The equation of "solution two" is

Then the projection area on the plane is

The surface of the two pieces of symmetry before and after, only consider one of the front, and then use the symmetry can be.

The two solutions of this example can be summed up the two main points of the surface area Division calculation:

1, give the correct equation form of the surface

Or

2, find the surface in the corresponding coordinate plane (or) on the projection area

(OR).

If the selection of surface equation is unsuitable, it will cause some difficulties for the determination of the projection area and the calculation of the double integral.

"Example 2" to find the coordinate of center of gravity of uniform surface.

Solution: Set surface density, center of gravity coordinate, according to the definition of center of gravity has

Which is the total mass of the surface,,, for the curved face of the coordinate plane, the moment.

The projection area of the surface in the surface is,

The area element of the surface is

So

By the symmetry of the surface, there are

，

Thus, the center of gravity coordinates.

The curvature area of the §10.5 coordinate

The projection area of the side and surface of a surface on the coordinate plane

Assuming we are talking about the surface is smooth, generally speaking, we encounter the surface are both sides, the surface can be defined by the surface of the normal vector to the point, this takes the normal vector also selected the side of the surface, we call it a directed surface.

is a curved surface that takes a small piece of surface to the cosine of the angle between the normal vector and the axis, and is the area value of the surface projection area. We stipulate that the projection on the plane is

Which is the case.

In short: The projection on the plane, the actual area on the surface of the projection area attached to a certain number, that is:.

Similarly can be defined on the surface and surface of the projection and.

Flow on one side of the surface flow

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

Given, it is a curved surface in the velocity field, and the function is continuous, and the mass of the fluid flowing to the specified side in the unit time is obtained, i.e. the flow rate.

The first is to discuss a special case: if the fluid flows through a closed area of the plane area, and the flow velocity of each point in the closed area is (constant vector), set to the plane of the unit normal vector.

Obviously, the fluid that flows through the closed area during the unit time consists of an oblique column with a bottom area of oblique height.

1, at that time, the volume of the oblique cylinder, which is through the closed area to the flow of the finger side;

2, at that time, it is clear that the flow through the closed region to the point of zero flows, and;

3. At that time, it means that the fluid flows through the closed area to the indicated side and flows to the indicated side. Therefore, regardless of the value, the flow of fluid through the closed region to the side of the direction of traffic is.

Then the general situation: the fluid flow is a surface, and the flow rate is variable, at this time the flow calculation can not directly use the above method, must use the element method to deal with.

Divide the surface into small pieces (and also represent the area of the small surface). If the diameter is small, we can use the flow rate at the point where it is smooth and continuous.

Instead of the flow rate at the other points,

Unit normal vector of a surface

Instead of the unit normal vector at each of the other points, the approximate value of the flow through the specified side of the flow is

So, by flowing to the specified side of the flow

But

So the upper-style can also be written

To take the limit of the above and the formula, the exact value of the flow is obtained.

Such limits will also be encountered in other problems, pumping out their specific meaning, can give the coordinates of the curvature of the concept of area.

The definition and properties of the curved Area division of the coordinate

The definition is set to a smooth, forward curved surface, and the function is bounded on. Arbitrarily divided into a small piece of the surface (at the same time also represents the area of a small piece of small surface), the projection on the plane is a point arbitrarily taken, if the size of the smaller surface diameter of the maximum value, the limit

Always exists, the limit is called the curvature area of the coordinate on the curved surface and is recorded.

That

Which is called the integrand, called the integral surface.

Similarly, you can define

The curvature area of a function on a curved surface, i.e.

The curvature area of a function on a curved surface, i.e.

The above three curved area points are also called the second kind of curved area.

We point out that when, in a continuous direction on a smooth surface, the curvature area of the coordinate is present, and is always assumed to be continuous in the future.

There are more forms in the application

For the sake of simplicity, we write it

For example, the flow above the specified side can represent a

If it is a piecewise smooth curved surface, we specify that the curvature of the function on the coordinates is equal to the curved area of the function on the smooth surface of each piece.

The surface integral of the coordinate has some properties similar to the curve integral of the coordinate.

1, if divided into and, then

(1)

The formula (1) can be extended to situations that are divided into several parts.

2, set is the direction of the surface, and to take the opposite side of the surface, then

(2)

(2) The formula shows that when the integral surface changes to the opposite side, the curved area of the coordinate should change the symbol, so we must pay attention to the side of the integral surface for the curvature area of the coordinate.

Proof of these properties Conlio.

The calculation method of the curved area division of the coordinate

The integral surface is the upper side of the surface given by the equation, the projection region on the plane, the assumption function has the first order continuous partial derivative on the surface, and is continuous on the upper part.

According to the definition of the curvature area of the coordinate, there

Because take the upper side, and so

And because it is