Question

计算（二重积分）xy^2dydz+yz^2dzdx+zx^2dxdy 范围为上半球面z=根号1-x^2-y^2的上侧

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.