$\frac{{\left({\mathrm{\rho }}^{2}U\right)}_{\mathrm{\rho }}}{{\mathrm{\rho }}^2}$

$\=\frac{{\partial }_{\mathrm{\rho }}\left({\mathrm{\rho }}^2\left(\frac{w \cos \left(\mathrm{\phi }\right)}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}\+\frac{w_{\mathrm{\phi }}}{\mathrm{\rho }}-\frac{v_{\mathrm{\θ}}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}\right)\right)}{{\mathrm{\rho }}^2}$

$\=\frac{{\partial }_{\mathrm{\rho }}\left(\mathrm{\rho } w \frac{\cos \left(\mathrm{\phi }\right)}{\sin \left(\mathrm{\phi }\right)}\+\mathrm{\rho } w_{\mathrm{\phi }}-\frac{\mathrm{\rho } v_{\mathrm{\θ}}}{\sin \left(\mathrm{\phi }\right)}\right)}{{\mathrm{\rho }}^2}$

$\=\frac{w\frac{\cos \left(\mathrm{\phi }\right)}{\sin \left(\mathrm{\phi }\right)}\+\mathrm{\rho } w_{\mathrm{\rho }}\frac{\cos \left(\mathrm{\phi }\right)}{\sin \left(\mathrm{\phi }\right)}\+w_{\mathrm{\phi }}\+\mathrm{\rho } w_{\mathrm{\phi } \mathrm{\rho }}-\frac{v_{\mathrm{\θ}}}{\sin \left(\mathrm{\phi }\right)}-\frac{\mathrm{\rho } v_{\mathrm{\θ} \mathrm{\ρ}}}{\sin \left(\mathrm{\phi }\right)} }{{\mathrm{\rho }}^2}$

$\=\frac{w}{{\mathrm{\rho }}^2}\frac{\cos \left(\mathrm{\phi }\right)}{\sin \left(\mathrm{\phi }\right)}\+\frac{w_{\mathrm{\rho }}}{\mathrm{\rho }}\frac{\cos \left(\mathrm{\phi }\right)}{\sin \left(\mathrm{\phi }\right)}\+\frac{w_{\mathrm{\phi }}}{{\mathrm{\rho }}^2}\+\frac{w_{\mathrm{\phi } \mathrm{\rho }}}{\mathrm{\rho }}-\frac{v_{\mathrm{\phi }}}{{\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)}-\frac{v_{\mathrm{\phi } \mathrm{\rho }}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}$

$\frac{{\left(V \sin \left(\mathrm{\phi }\right)\right)}_{\mathrm{\phi }}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}$

$\=\frac{{\partial }_{\mathrm{\phi }}\left(\left(\frac{u_{\mathrm{\theta }}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}-\frac{w}{\mathrm{\rho }}-w_{\mathrm{\rho }}\right)\sin \left(\mathrm{\phi }\right)\right)}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}c$

$\=\frac{{\partial }_{\mathrm{\phi }}\left(\frac{u_{\mathrm{\theta }}}{\mathrm{\rho }}-\frac{w \sin \left(\mathrm{\phi }\right)}{\mathrm{\rho }}-w_{\mathrm{\rho }}\sin \left(\mathrm{\phi }\right)\right)}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}$

$\=\frac{\frac{u_{\mathrm{\theta } \mathrm{\phi }}}{\mathrm{\rho }}-\frac{w_{\mathrm{\phi }} \sin \left(\mathrm{\phi }\right)\+w \cos \left(\mathrm{\phi }\right)}{\mathrm{\rho }}-w_{\mathrm{\rho }}\cos \left(\mathrm{\phi }\right)-w_{\mathrm{\rho } \mathrm{\phi }}\sin \left(\mathrm{\phi }\right)}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}$

$\=\frac{u_{\mathrm{\θ} \mathrm{\φ}}}{{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)}-\frac{w_{\mathrm{\φ}}}{{\mathrm{\ρ}}^2}-\frac{w}{{\mathrm{\ρ}}^2}\frac{\cos \left(\mathrm{\φ}\right)}{\sin \left(\mathrm{\φ}\right)}-\frac{w_{\mathrm{\ρ}}}{\mathrm{\ρ}}\frac{\cos \left(\mathrm{\φ}\right)}{\sin \left(\mathrm{\φ}\right)}-\frac{w_{\mathrm{\ρ} \mathrm{\φ}}}{\mathrm{\ρ}}$

$\frac{W_{\mathrm{\theta }}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}$

$\=\frac{{\partial }_{\mathrm{\theta }}\left(\frac{v}{\mathrm{\rho }}\+v_{\mathrm{\rho }}-\frac{u_{\mathrm{\phi }}}{\mathrm{\rho }}\right)}{ \mathrm{\rho } \sin \left(\mathrm{\phi }\right)}$

$\=\frac{\frac{v_{\mathrm{\theta }}}{\mathrm{\rho }}\+v_{\mathrm{\rho } \mathrm{\theta }}-\frac{u_{\mathrm{\phi } \mathrm{\theta }}}{\mathrm{\rho }}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}$

$\=\frac{v_{\mathrm{\θ}}}{{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)}\+\frac{v_{\mathrm{\ρ} \mathrm{\θ}}}{\mathrm{\ρ} \sin \left(\mathrm{\φ}\right)}-\frac{u_{\mathrm{\φ} \mathrm{\θ}}}{{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)}$

Table 9.4.5(a)   Divergence of the curl in spherical coordinates

$\frac{w}{{\mathrm{\rho }}^2}\frac{\cos \left(\mathrm{\phi }\right)}{\sin \left(\mathrm{\phi }\right)}\+\frac{w_{\mathrm{\rho }}}{\mathrm{\rho }}\frac{\cos \left(\mathrm{\phi }\right)}{\sin \left(\mathrm{\phi }\right)}\+\frac{w_{\mathrm{\phi }}}{{\mathrm{\rho }}^2}\+\frac{w_{\mathrm{\phi } \mathrm{\rho }}}{\mathrm{\rho }}-\frac{v_{\mathrm{\θ}}}{{\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)}-\frac{v_{\mathrm{\θ} \mathrm{\ρ}}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}$

$\frac{u_{\mathrm{\theta } \mathrm{\phi }}}{{\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)}-\frac{w_{\mathrm{\phi }}}{{\mathrm{\rho }}^2}-\frac{w}{{\mathrm{\rho }}^2}\frac{\cos \left(\mathrm{\phi }\right)}{\sin \left(\mathrm{\phi }\right)}-\frac{w_{\mathrm{\rho }}}{\mathrm{\rho }}\frac{\cos \left(\mathrm{\phi }\right)}{\sin \left(\mathrm{\phi }\right)}-\frac{w_{\mathrm{\rho } \mathrm{\phi }}}{\mathrm{\ρ}}$

$\frac{v_{\mathrm{\theta }}}{{\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)}\+\frac{v_{\mathrm{\rho } \mathrm{\theta }}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}-\frac{u_{\mathrm{\phi } \mathrm{\theta }}}{{\mathrm{\rho }}^2\sin \left(\mathrm{\phi }\right)}$

Table 9.4.5(b)   The terms in $\nabla ·\left(\nabla \times \mathbf{F}\right)$ that are to be combined

Initialize

Loading Student:-VectorCalculus

Define the spherical vector field F

$⟨u\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,v\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,w\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)⟩$ = $\left[\begin{array}{c}u\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\\ v\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\\ w\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\end{array}\right]$$\stackrel{\text{apply coordinates}}{\to }$$\left[\begin{array}{c}u\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\\ v\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\\ w\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\end{array}\right]$$\stackrel{\text{to Vector Field}}{\to }$$\left[\begin{array}{c}u\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\\ v\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\\ w\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$F$

Compute the divergence of the curl of F

$\nabla ·\left(\nabla \times \mathbf{F}\right)$ = $0$$$

Initialize

$$\begin{gathered} \mathrm{with}\left(\mathrm{Student}:-\mathrm{VectorCalculus}\right)\: \\ \mathrm{BasisFormat}\left(\mathrm{false}\right)\: \end{gathered}$$

Implement notational simplifications with the declare command in the PDEtools package

$\mathrm{PDEtools}:-\mathrm{declare}\left(u\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,v\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,w\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,\mathrm{quiet}\right)$

Use the VectorField command in the Student VectorCalculus package to define F

$\mathbf{F}≔\mathrm{VectorField}\left(⟨u\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,v\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,w\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)⟩\,\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\right)$ = $$

Verify $\nabla ·\left(\nabla \times \mathbf{F}\right)\=0$

$\mathrm{Divergence}\left(\mathrm{Curl}\left(\mathbf{F}\right)\right)$ = $0$$$