§ 4 Area functions

 

1. Area function and density function

 

       Let σ be the area of ​​space Ω , when the variable u changes with the change of the area σ, we call u a function of the area σ , denoted as u = u ( σ ). Let the measure of the area (length, area or volume) still be recorded as Let σ , subdivide σ infinitely so that the measure of the small region containing M (a point in the region) → 0 , if u changes by Δ u , and the limit

                    

exists, then u ( M ) is called the density of the area function u ( σ ) , denoted as

                    

It is a function of point M , called the density function, and its differential form is

                                                                            (1)

       For example, when σ is a surface in three-dimensional space, ( M ) is the surface density of the surface, and when σ is an object in three-dimensional space, ( M ) is the volume density . In fact, ( M ) is all x, y, z three-variable function .

 

Second, the integral of the density function

 

       is obtained from (1)

                    

Due to the change of the integral area, these integrals are again various forms of area functions, such as

       [ Linear density and single integral on a straight line ]  Let σ = I = [ a, b ], then the mass of the non-uniform thin rod

             

where is the linear density .

       [ Area Density and Double Integral on a Plane ]   Let the density of the sheet be ( x, y ) , then the mass of the sheet is

                    

where D represents the area of ​​the sheet on the plane .

       [ Volume Density and Triple Integral ]   Let the density of the object be ( x , y, z ) , then the mass of the object is

                    

where V represents the area of ​​the object in space .

       [ Integration of linear density with respect to arc length element ]   Since the mass element of curve segment l is

                    

So the mass of the curve segment is

                    

       The mass of the [ area density on the surface and the integral with respect to the surface elements ]   distributed on the surface S is

                    

where D is the projected area of ​​S on the Oxy plane, and the equation of S is z = f ( x ,y ), x,y ∈ D . The integral of this form is called the integral of the function ( M ) with respect to the surface element d S .

 

3. The concept of delta - function

 

       When discussing the distribution of concentrated mass on a straight line, for example, there is only a concentrated mass m at the origin x = 0 , and there is no mass at other points, at this time, for the point on the straight line whose coordinates are not equal to zero, it is always possible to take the point that contains this point A sufficiently small interval Δ l , so that Δ l does not include the point x = 0 , then the average density on the interval Δ l is zero, so the density at the point x ( ≠ 0) is also zero; and at the point x = 0 , If Δl contains this point, since

               

When Δ l → 0 , the average density tends to infinity . In any interval [ a, b ] containing the point x = 0 , the mass is always equal to m , and the density function u (x) is still considered at this time,

                    

                    

In particular, when m = 1 , denote this density function as δ ( x ) , that is

                     \

For any interval I containing the point x = 0 , we have

                or 

Call this function δ ( x ) a δ - function ( also called a Dirac function or an impulse function ).

       The δ - function has an important property: for any continuous function f ( x ), we have

             

This property shows that although the δ - function does not conform to the classical function definition of "one point corresponds to a point", the integral of the product of it and any continuous function within ( -∞ ,∞) has a clear definition . Therefore , the δ - function is in It has a wide range of applications in modern physics and engineering technology .