Question

_r/_x怎么求

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 sin_φ - r sinθ cos_φ) - (r cosθ sinφ)(- r cosθ sin_φ - r cosθ cos_φ)
= (- r sinθ sinφ)(- r sinθ) - (r cosθ sinφ)(- r cosθ)
= r_ sin_θ sinφ + r_ cos_θ sinφ
= r_ sinφ
——————————————————————————————————————————
=> dxdydz = | J | dr dφ dθ = r_ sinφ dr dφ dθ
——————————————————————————————————————————
于是x_ + y_ + z_
= (r_ cos_θ sin_φ + r_ sin_θ sin_φ) + r_ cos_φ
= r_ sin_φ + r_ cos_φ
= r_
——————————————————————————————————————————
∫∫∫Ω (x_ + y_ + z_) dx dy dz
= ∫∫∫Ω r_ * r_ sinφ dr dφ dθ
= ∫[0→2π] dθ ∫[0→π] sinφ dφ ∫[0→r] r_ dr
= 2π * 2 * r_/5
= (4/5)πr_