7)Криволинейный интеграл и его механический смысл. X=x(t)

7) . X=x(t) Y=y(t) . V() AoAn T= Ao(x(); y()) T= An(x(); y()) X=R cost Y=Rsint (;) F - - F(x;y)=P(x;y)i+Q(x;y)j A -? . T [;] V AoA1,A1A2.An-1,An [;] [to;t1]=t*1 [tn-1;tn]=t*n . A*I (x(t*I);y(t*I)) F*(x(t*I);y(t*I)) Ai-1.Ai max [Ai-1;Ai] n Lim F*(x(t*I);y(t*I)) Ai-1.Ai , n A*I AoAn F(x;y)dx+Q(x;y)dy F,Q . Fdr, dr- F (- ) A=Fdr AoAn P(x;y) Q(x;y) P(x;y)dx+Q(x;y)dy 8) . . , , - . . L D/ || . 2 . Pdx+Qdy = (dQ\dx dP\dy)dxdy -. dP\dy dxdy=dx(dP\dy dy=dx P(x;y) 1 2 = dx[x;2(x))-P(x;1(x)) Pdx=-(dP\dy ) dy dQ\dy . 9) . Pdx+Qdy ( ) = 0 . Pdx+Qdy=0 (AmB) Pdx+Qdy=(AnB)Pdx+Qdy 0=Pdx+Qdy=(AnB)Pdx+Qdy+ (BnA)Pdx+Qdy=(AnB) - (AnB . : (AmB) Pdx+Qdy=(AnB)Pdx+Qdy . Pdx+Qdy=0 (AmB) Pdx+Qdy=(AnB)Pdx+Qdy, (AmB) Pdx+Qdy-(AnB)Pdx+Qdy,=

(AnB) Pdx+Qdy+(BnA)Pdx+Qdy=0 (AnB) Pdx+Qdy+(BnA)Pdx+Qdy= (AmBnA)Pdx+Qdy Pdx+Qdy dP\dy=dQ\dx Pdx+Qdy=(dP\dy-dQ\dx)dxdy= 0dxdy=0. 1. =0 2. . 3. dP\dy=dQ\dX (c) 10) . .. . {an} .a1 + a2+ a3 + + an=Ean , . . u1 + u2 + u3 + ... + un + ... , un ( ) :Lim Un=0 ( !). . 11) . . . {an} , q, . , . , q 0 q 1. : n

1. Ean S, Ec * an c * S . An :Ean An= a1+a2++an , Sn- E c*an Sn= c*a1 + c*a2 ++c*an=c(a1+a2+ an)=c*An n Lim Sn=Lim(c*An)=c*S. E c*an = c*S,. Limc*an=c*S 2. Ean Ebn , S1 S2, E(an+-bn) .S1+S2 . Ean,Ebn, Sn E(an+-bn), Sn=(a1+-b1)+(a2+-b2)+.+(an+bn)=(a1+a2+..an)+-(b1+b2+bn)=S^an+S^bn Sn n LimSn=lim(S^an+-S^bn)=lim(Sn)^a+lim(Sn)^b =S1+-S2 . . Sn E(an+-bn) S1+-S2 E(an+bn) . 3. . . Sn n- (1), Ck (, n Sn), sn-k , Sn Ck. : Ck , n. , LimSn , LimSn ,

-. , an=f(n), 1 f(x)dx. , , . . f(x)

. , Ean, (n+1)- , , n- :

, .

. . , | q| < 1, , | q| > 1. , , Sn = 1 + 1/2 + 1/3 + ... + 1/n Sn ln n : lim (Sn ln n) = C n 12 - . .

=ak+1 + ak+2+... , , .. .Lim r n =0 13) . 1) 2 an bn. n=N, N( n>= N) an= 1 f(x)dx 1 f(x)dx = (. ), lim Sn= . an=1/n^p , >1 , 1, p,