证明令=x+i△x=x1-x1△pyk=yk-y1 Sk= 5k tink u(sk, nk=uk v(sk, nk)=vk Sn=∑f(5k△k=2(+i)△x+边 k=1 ∑以(5s,mx=∑v,mn 人k当δ→0时,均是 k=1 k=1 实函数的曲线积分 +∑v9,mAx+∑(5,m)A(5 Sky k=1 limS,=lim >f(k)Azk=( u(x,y)dx- v(x, y) dy) n→0 n1→0 +订vx,m(x,yh)=∫/(mhk 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