Шпора на экзамен english version (2 курс английской группы МО) по лекциям Усмановой / Curvilinear integral of the first kind

1. Curvilinear integral of the first kind. Definition.

At any point linear density=ρ(µ). Our task is to find the mass of all curve. Lets divide it into parts: A[1], A[2]..A[n-1].

A[0]=A, A[n]=B. In each arch lets take point µ[i]. Aproximately supposing that density at all archs is the same as at point µ[i]. Mass of the µ[i] is m[i]= ρ(µ[i])*σ[i] (σ[i]lenght of the arch).

The error of this expression will be approached to zerro when all σ[i] will be approach to zero. Lets denote the largest lenght λ=max(σ[i]).

(*)

lets investigate limits of the such kind (*) in total. Suppose we have some function of point f(μ)=f(x,y) defined the along continous curve k and lets repeat the process. Lets divide k into elementary archs, then lets choose point µ[i] in each arch (**), This expression will be called integral sum in a set.

Def.: If (**) has defined finite limit I, independent neither of dividing way nor of choosing points, when lenght approaches to zero , then it called curvilinear integral of the first kind of function taken along the curve k.

The curvilinear integral of the first kind doesnt depend on the direction of the curve. Mass of the curve is

2. Reduction to definite integral

Position of M could be defined by arch length S= AM, where A is inition point. That we express curve K by parametric equation x=x(s), y=y(s) (0<=s<=S)

f=f(x(s),y(s)). Lets denote the arch lenght S[i] responding to dividing points A[i]. Then lenght of one arch A[i]A[i+1] will be σ[i]=s[i+1]-s[i]< and lets denote s[i]* µ[i] (the length of s*=Aµ[i]).

(1.3)

Let our curve may be expressed by parametric expression x=φ(t), y=ψ(t).

Functions ψ,φ may have continous derivatives φ’(t), ψ’(t).

If we defined K in explicit form (y=y(x)), then:

3. Curvilinear integral of the second kind. Definition

f(μ)=f(x,y)

Divide the curve into parts μ[i](ξ[i],η[i])

Lets compose an integral sum

Def: if this sum has finite limit I, independent neither of dividing way nor of of choosing points, when μ=max(A[i],A[i+1]) approaches to zero, then it called curvilinear integral of the second kind of function f(M)dx

If we multiply function value onto projection of axis Oy, then

4. Existence and calculation of the curv integral of 2 kind

Let curve K be defined by parametric equation x=φ(t), y=ψ(t) (α<=t<=β)

Functions φ,ψ are continous. Also we suppose that changing parameter from α to β, result in describing curve just from A to B. Function f(x,y) also sup[pose to be continous and if we take projrction onto Ox, we should supplementary to preset existence and continously of φ’(t).

Rule: under this condition integral

exists and can be calculated by the formular:

Proof: Let point A[i] corresponds to parameter various t[i]. Then integral sum