令P=2x,Q=yz,R=-z²
∵αP/αx=2,αQ/αy=z,αR/αz=-2z
∴根据高斯公式得
原式=∫∫∫(αP/αx+αQ/αy+αR/αz)dxdydz (V是S围城的空间区域)
=∫∫∫(2-z)dxdydz
=∫dθ∫rdr∫(2-z)dz (应用柱面坐标变换)
=2π∫[2r√(2-r²)-r-2r²+r³]dr
=2π[(-2/3)(2-r²)^(3/2)-r²/2-(2/3)r³+r^4/4]│
=2π(-1/2-2/3+1/4+2/3)
=-π/2.