題目

Problem

Part I Multiple Choice

Consider the spherical coordinates system $x=\rho\sin\phi\cos\theta$ , $y=\rho\sin\phi\sin\theta$ , and $z=\rho\cos\phi$ . The volume element $\mathrm{d}V=\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$ can be changed as

(A) $\rho\sin\phi\,\mathrm{d}\rho\,\mathrm{d}\theta\,\mathrm{d}\phi$ .

(B) $\rho\cos\phi\,\mathrm{d}\rho\,\mathrm{d}\theta\,\mathrm{d}\phi$ .

(D) $\rho^2\cos\phi\,\mathrm{d}\rho\,\mathrm{d}\theta\,\mathrm{d}\phi$ .

解答

展開

展開

對

x=\rho\sin\phi\cos\theta, \quad y=\rho\sin\phi\sin\theta, \quad z=\rho\cos\phi,

其 Jacobian 的絕對值為

\left|\frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}\right|=\rho^2\sin\phi.

因此

\mathrm{d}V=\rho^2\sin\phi\,\mathrm{d}\rho\,\mathrm{d}\theta\,\mathrm{d}\phi,

故正確選項為 $\boxed{\text{C}}$ 。