中南大学：《大学数学》课程PPT教学课件（微积分案例题解）chapter4（8）曲线积分与曲面积分小结

4对弧长的曲线积分的应用: K心[ (1)当P(x,y)表示L的线蜜度时,m=』p(x,yd (2)当f(x,y)≡时,I弧长=d; roas (3)曲线弧的重心坐标x= ipds pds (4)曲线弧的转动惯量 Ix=fy'ads, I,=Leads, Io=L(x2+y2)Ads 对空间曲线构件也有结论!

(1) 当(x, y)表示L的线密度时, ( , ) ; = L m  x y ds (2) ( , ) 1 , ;  = L 当 f x y 时 L弧长 ds (3) 曲线弧的重心坐标 , .     = = L L L L ds y ds y ds x ds x     (4) 曲线弧的转动惯量 , , ( ) . 2 2 2 2 =  =  =  + L o L y L I x y ds I x ds I x y ds 对空间曲线构件也有结论! 4.对弧长的曲线积分的应用:

对坐标的曲线积分 1定义:∫P(x,y)d=lm∑P(5,m)Ax -> 0 ∫Q(x,y)=lim∑Q(5,n)4 -) P(x,],a)dx=lim >P(Si, ni, si)Ax; 九- 0 i=1 Q(, y, z)dy=lim ∑Q(,n,)An 入->0 R(x,y,)z=lim∑R(,n,,)△z 入→>0 K心

二. 对坐标的曲线积分 1.定义: ( , ) lim ( , ) 1 0   → = =  n i i i i L P x y dx P   x  ( , ) lim ( , ) 1 0   → = =  n i i i i L Q x y dy Q   y  ( , , ) lim ( , , ) . 1 0 i i i n i P x y z dx = P i x  =  →     ( , , ) lim ( , , ) . 1 0 i i i n i i Q x y z dy = Q    y  =  → ( , , ) lim ( , , ) . 1 0 i i i n i i R x y z dz = R    z  =  →

2对坐标的曲线积分的性质: (1J,KP(r,Ddx+k22(x,])dy kIL, P(x, y)dx+K2S,o(x,y)dy (2)如果把L分成L1和L2,则 「p=J,P+Q+[,Pd+g小 (3)JP(x,y)+(xy)=-P(x,y)dk+q(x,y 即对坐标的曲线积分与曲线的方向有关 K心

2.对坐标的曲线积分的性质: ( , ) ( , ) . (1) ( , ) ( , ) 1 2 1 2    = + + L L L k P x y dx k Q x y dy k P x y dx k Q x y dy . (2) , 1 2 1 2  + =  + +  + L L L Pdx Qdy Pdx Qdy Pdx Qdy 如果把L分成L 和L 则  + = − + −L L (3) P(x, y)dx Q(x, y)dy P(x, y)dx Q(x, y)dy 即对坐标的曲线积分与曲线的方向有关

3对坐标的曲线积分的计算: (1)直接计算法:方法:一代二换三定限 fPdr+edy x=o(t) y=y(t tma-piplo y=y(x Pac+二「m (Pl,y(x)]+elx, y(xly(x)be 从 → b Px+Q小y J从c→de (PIo(y), ylo(y)+elp(y), ylldy K心

3.对坐标的曲线积分的计算: (1) 直接计算法: 方法：一代二换三定限  { [ ( ), ( )] ( ) [ ( ), ( )] ( )} . ( ) ( ) P t t t Q t t t dt Pdx Qdy y t x t t L   =====  +  + = = →             从 { [ , ( )] [ , ( )] ( )} . ( ) Pdx Qdy P x x Q x x x dx b a y x x a b L  + ===== +  = →     从 { [ ( ), ] ( ) [ ( ), ]} . ( ) Pdx Qdy P y y y Q y y dy d c x y y c d L + =====   + = →     从

r(e)cos ly=r(O) 8 ∫Pd+Q小 d 从a→月a FPac+g小+Rh P y=y z=o(t) (PIo(),(t),a(t)lo'(t t从a→B + elo(t),y(t), a(tly'(t rIo(t),(t),a(tla(tft K心

 . ( )sin ( )cos           Pdx Qdy d y r x r L  = = → + ========  从 R t t t t dt Q t t t t P t t t t Pdx Qdy Rdz z t y t x t t [ ( ), ( ), ( )] ( )} [ ( ), ( ), ( )] ( ) { [ ( ), ( ), ( )] ( ) ( ) ( ) ( )                    +  +  =====  + +       = = = →  从