Question

曲线曲面积分题

Answer 1

设P=yz(x-x^2)

 Q=zx(y-y^2)

R=xy(z-z^2)

所以根据斯托克斯公式

原积分

=∫Pdx+Qdy+Rdz

=

=∫∫∑    [x(z-z^2)-x(y-y^2)]dydz+[y(x-x^2)-y(z-z^2)]dzdx+[z(y-y^2)-z(x-x^2)]dxdy

(因为∑在x+y+z=0上，所以dydz: dzdx: dxdy=1:1:1)

=∫∫∑1   [x(z-z^2)-x(y-y^2)+y(x-x^2)-y(z-z^2)+z(y-y^2)-z(x-x^2)]dxdy

=∫∫∑1    (-xz^2+xy^2-yx^2+yz^2-zy^2+zx^2)dxdy

=∫∫∑1    (3xy^2-3x^2y+2y^3-2x^3)dxdy

=∫∫∑1    f(x,y)dxdy

(∑1为椭圆2x^2+2xy+2y^2=a^2,  他是关于原点对称的积分区域）

设f(x,y)=3xy^2-3x^2y+2y^3-2x^3

因为满足f(-x,-y)=-f(x,y)

所以∫∫f(x,y)dxdy=0

所以

原积分=0