$\nabla ·\left(f \mathbf{F}\right)$

$\=\frac{{\left({\mathrm{\rho }}^{2}u f\right)}_{\mathrm{\rho }}}{{\mathrm{\rho }}^2}\+\frac{{\left(v f \sin \left(\mathrm{\phi }\right)\right)}_{\mathrm{\phi }}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}\+\frac{{\left(w f\right)}_{\mathrm{\theta }}}{\mathrm{\rho } \sin \left(\mathrm{\phi }\right)}$

$\=\frac{2fu}{\mathrm{\rho }}\+f{u}_{\mathrm{\ρ}}\+\frac{f\cos \left(\mathrm{\phi }\right)v}{\mathrm{\rho }\sin \left(\mathrm{\phi }\right)}\+\frac{f v_{\mathrm{\φ}}}{\mathrm{\rho }}\+\frac{f w_{\mathrm{\θ}}}{\mathrm{\rho }\sin \left(\mathrm{\phi }\right)}\+u{f}_{\mathrm{\ρ}}\+\frac{v{f}_{\mathrm{\φ}}}{\mathrm{\rho }}\+\frac{w{f}_{\mathrm{\θ}}}{\mathrm{\rho }\sin \left(\mathrm{\phi }\right)}$

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Additional notational devices

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left\{f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,u\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,v\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,w\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\right\}\right)$

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

Define the vector field F

$⟨u\,v\,w⟩$ = $\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$$\stackrel{\text{apply coordinates}}{\to }$$\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$$\stackrel{\text{to Vector Field}}{\to }$$\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$$\stackrel{\text{assign to a name}}{\to }$$F$$$

Implement the left side of Identity 5: $\nabla ·\left(f \mathbf{F}\right)\=f \nabla ·\mathbf{F}\+\mathbf{F}·\left(\nabla f\right)$ 

$\nabla ·\left(f \mathbf{F}\right)$

$\frac{2\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)fu\+{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)f_{\mathrm{\rho }}u\+{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)f{u}_{\mathrm{\rho }}\+\mathrm{\ρ}\cos \left(\mathrm{\φ}\right)fv\+\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)f_{\mathrm{\phi }}v\+\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)f{v}_{\mathrm{\phi }}\+\mathrm{\ρ}{f}_{\mathrm{\theta }}w\+\mathrm{\ρ}f{w}_{\mathrm{\theta }}}{{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)}$

$\stackrel{\text{expand}}{\=}$

$\frac{2fu}{\mathrm{\ρ}}\+f{u}_{\mathrm{\rho }}\+\frac{f\cos \left(\mathrm{\φ}\right)v}{\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)}\+\frac{f{v}_{\mathrm{\phi }}}{\mathrm{\ρ}}\+\frac{f{w}_{\mathrm{\theta }}}{\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)}\+u{f}_{\mathrm{\rho }}\+\frac{v{f}_{\mathrm{\phi }}}{\mathrm{\ρ}}\+\frac{w{f}_{\mathrm{\theta }}}{\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)}$

Implement the right side of Identity 5: $\nabla ·\left(f \mathbf{F}\right)\=f \nabla ·\mathbf{F}\+\mathbf{F}·\left(\nabla f\right)$ 

$f \left(\nabla ·\mathbf{F}\right)\mathbf{\+}\mathbf{F}·\mathrm{Gradient}\left(f\,\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\right)$

$\frac{f\left(2\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)u\+{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)u_{\mathrm{\rho }}\+\mathrm{\ρ}\cos \left(\mathrm{\φ}\right)v\+\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)v_{\mathrm{\phi }}\+\mathrm{\ρ}{w}_{\mathrm{\theta }}\right)}{{\mathrm{\ρ}}^2\sin \left(\mathrm{\φ}\right)}\+u{f}_{\mathrm{\rho }}\+\frac{v{f}_{\mathrm{\phi }}}{\mathrm{\ρ}}\+\frac{w{f}_{\mathrm{\theta }}}{\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)}$

$\stackrel{\text{expand}}{\=}$

$\frac{2fu}{\mathrm{\ρ}}\+f{u}_{\mathrm{\rho }}\+\frac{f\cos \left(\mathrm{\φ}\right)v}{\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)}\+\frac{f{v}_{\mathrm{\phi }}}{\mathrm{\ρ}}\+\frac{f{w}_{\mathrm{\theta }}}{\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)}\+u{f}_{\mathrm{\rho }}\+\frac{v{f}_{\mathrm{\phi }}}{\mathrm{\ρ}}\+\frac{w{f}_{\mathrm{\theta }}}{\mathrm{\ρ}\sin \left(\mathrm{\φ}\right)}$