82-sc-r4
从笛卡尔坐标 (x,y,z)(x, y, z)(x,y,z) 到柱面坐标 (r,θ,z)(r, \theta, z)(r,θ,z) 的变换由以下关系给出：
det(J)=det[cosθ−rsinθ0sinθrcosθ0001]=r(cos2θ+sin2θ)=r\det(J) = \det\begin{bmatrix} \cos \theta & -r \sin \theta & 0 \\ \sin \theta & r \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} = r (\cos^2 \theta + \sin^2 \theta) = r det(J)=detcosθsinθ0−rsinθrcosθ0001=r(cos2θ+sin2θ)=r
因此，新的体积元素为 dV=r dr dθ dzdV = r \, dr \, d\theta \, dzdV=rdrdθdz。
∫02πdθ∫0cr⋅dr∫0(rcosθa)2+(rcosθa)2dz=c4π8\int_0^{2\pi} d\theta \int_0^c r\cdot dr \int_0^{\Big(\dfrac{r\cos\theta}{a}\Big)^2+\Big(\dfrac{r\cos\theta}{a}\Big)^2}dz =\frac{c^4\pi}{8} ∫02πdθ∫0cr⋅dr∫0(arcosθ)2+(arcosθ)2dz=8c4π
秒了
