Question

二重积分求球表面

Answer 1

令P=xy2,Q=yz2,R=zx2

则αP/αx=y2,αQ/αy=z2,αR/αz=x2

∴根据高斯定理,有

∫∫xy2dydz+yz2dzdx+zx2dxdy+∫∫xy2dydz+yz2dzdx+zx2dxdy

=∫∫∫(αP/αx+αQ/αy+αR/αz)dxdydz

=∫∫∫(x2+y2+z2)dxdydz (D表示上半球面,S表示xy平面圆：x2+y2=1,V表示D+S)

=∫dθ∫sinφdφ∫r2*r2dr (做球面坐标变换)

=(2π-0)(1-0)(1/5-0)

=2π/5

∵∫∫xy2dydz+yz2dzdx+zx2dxdy=0 (∵z=0,∴dz=0)

∴∫∫xy2dydz+yz2dzdx+zx2dxdy=2π/5-∫∫xy2dydz+yz2dzdx+zx2dxdy

=2π/5-0

=2π/5.