证明令zk=x4+i△x=xk-x21Ay=y-yk1 k t+iu(们k)=h以(,n1)=v k n S=∑f(4)Ak=∑(ak+ik)△xk+边yk) k=1 ∑m(5,m)=∑ v(,、D当δ→Q时,均是 =1 =1 实函数的曲线积分 +2(5k,k)Axk+∑(5k,m)(5) =1 imSn=im∑f(k)△x=((x,y)-「v(x,p)y) n→ +i(v(x,y)dx+u(x, y)dy )=f(zdzk k k k k k k k k k k k k k k k k k i u u v v z x i y x x x y y y = + = = = +  = − −  = − − ( , ) ( , ) 1 1        令 [ ( , ) ( , ) ] (5) ( , ) ( , ) 1 1 1 1     = = = = +  +  =  −  n k k k k n k k k k n k k k k n k k k k i v x u y u x v y           = = =  = +  +  n k k k k k n k n k k S f z u i v x i y 1 1 ( ) ( )( )       + + =  =  = − = → → C C C C C n k k k n n n i v x y dx u x y dy f z dz S f z u x y dx v x y dy ( ( , ) ( , ) ) ( ) lim lim ( ) ( ( , ) ( , ) ) 1  证明 . 0 实函数的曲线积分 当 → 时，均是