Question

计算曲面积分∫∫x^3dydz+y^3dzdx+z^3dxdy,∑是上半球面z=根下1-x^2-y^2的上侧

Answer 1

简单分析一下，答案如图所示

Answer 2

在半球面∑上添加圆面S：(x²+y²=1,z=0),使之构成封闭曲面V=∑+S.
∵∫∫x³dydz+y³dzdx+z³dxdy=0 (∵z=0,∴dz=0)
∴ ∫∫x³dydz+y³dzdx+z³dxdy+∫∫x³dydz+y³dzdx+z³dxdy
=∫∫∫(3x²+3y²+3z²)dxdydz (应用高斯公式)
=3∫∫∫(x²+y²+z²)dxdydz
=3∫dθ∫dφ∫r²*r²sinφdr (作球面坐标变换)
=3*(2π)*(cos(0)-cos(π/2))*(1^5/5-0^5/5)
=6π/5
故∫∫x³dydz+y³dzdx+z³dxdy=∫∫∫(3x²+3y²+3z²)dxdydz-∫∫x³dydz+y³dzdx+z³dxdy
=6π/5-0
=6π/5.