球体曲面积分x平方+y平方+z平方=r平方用高斯解中间有一步∫∫∫(x²+y²+z²)dxdydz=∫∫∫r^2*(r^2sinφ)drdφdθ麻烦写下这步中间具体过程

Question

球体曲面积分 x平方+y平方+z平方=r平方
用高斯解 中间有一步∫∫∫(x²+y²+z²)dxdydz=∫∫∫r^2*(r^2sinφ)drdφdθ 麻烦写下这步中间具体过程

Answer 1

运用球面坐标、设：
x = r cosθ sinφ
y = r sinθ sinφ
z = r cosφ
=> ∂x/∂r = cosθ sinφ、∂x/∂φ = r cosθ cosφ、∂x/∂θ = - r sinθ sinφ
=> ∂y/∂r = sinθ sinφ、∂y/∂φ = r sinθ cosφ、∂y/∂θ = r cosθ sinφ
=> ∂z/∂r = cosφ、∂z/∂φ = - r sinφ、∂z/∂θ = 0
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J = ∂(x,y,z)/∂(r,φ,θ)
= | ∂x/∂r | ∂x/∂φ | ∂x/∂θ |
= | ∂y/∂r | ∂y/∂φ | ∂y/∂θ |
= | ∂z/∂r | ∂z/∂φ | ∂z/∂θ |
——————————————————————————————————————————
= | cosθ sinφ | r cosθ cosφ | - r sinθ sinφ |
= | sinθ sinφ | r sinθ cosφ | r cosθ sinφ |
= | cosφ | - r sinφ | 0 |
——————————————————————————————————————————
= (- r sinθ sinφ)[ (sinθ sinφ)(- r sinφ) - (r sinθ cosφ)(cosφ)]
- (r cosθ sinφ)[ (cosθ sinφ)(- r sinφ) - (r cosθ cosφ)(cosφ) ]
= (- r sinθ sinφ)(- r sinθ sin²φ - r sinθ cos²φ) - (r cosθ sinφ)(- r cosθ sin²φ - r cosθ cos²φ)
= (- r sinθ sinφ)(- r sinθ) - (r cosθ sinφ)(- r cosθ)
= r² sin²θ sinφ + r² cos²θ sinφ
= r² sinφ
——————————————————————————————————————————
=> dxdydz = | J | dr dφ dθ = r² sinφ dr dφ dθ
——————————————————————————————————————————
于是x² + y² + z²
= (r² cos²θ sin²φ + r² sin²θ sin²φ) + r² cos²φ
= r² sin²φ + r² cos²φ
= r²
——————————————————————————————————————————
∫∫∫Ω (x² + y² + z²) dx dy dz
= ∫∫∫Ω r² * r² sinφ dr dφ dθ
= ∫[0→2π] dθ ∫[0→π] sinφ dφ ∫[0→r] r⁴ dr
= 2π * 2 * r⁵/5
= (4/5)πr⁵