曲面积分 ,补上平面∑1:z=0下侧,利用高斯公式:
∫∫x^2dydz+y^2dxdz+z^2dxdy
=2∫∫∫(x+y+z)dxdydz-∫∫[∑1]x^2dydz+y^2dxdz+z^2dxdy
=2∫∫∫zdxdydz
=2π∫[0,1](1-z^2)zdz
=-1/2π(1-z^2)^2|[0,1]
=π/2
曲面积分 ,补上平面∑1:z=0下侧,利用高斯公式:
∫∫x^2dydz+y^2dxdz+z^2dxdy
=2∫∫∫(x+y+z)dxdydz-∫∫[∑1]x^2dydz+y^2dxdz+z^2dxdy
=2∫∫∫zdxdydz
=2π∫[0,1](1-z^2)zdz
=-1/2π(1-z^2)^2|[0,1]
=π/2