$\frac{{\left(r U\right)}_r}{r}\+\frac{V_{\mathrm{\theta }}}{r}\+W_z$

$\=\frac{{\partial }_r\left(r \left(\frac{w_{\mathrm{\theta }}-{\left(r v\right)}_z}{r}\right)\right)}{r}\+\frac{{\partial }_{\mathrm{\θ}}\left(u_z-w_r\right)}{r}\+{\partial }_z\left(\frac{{\left(r v\right)}_r-u_{\mathrm{\theta }}}{r}\right)$

$\=\frac{{\partial }_r\left(w_{\mathrm{\θ}}-r v_z\right)}{r}\+\frac{u_{z\mathrm{\θ}}-w_{r\mathrm{\θ}}}{r}\+\frac{{\partial }_z\left(v\+r v_r-u_{\mathrm{\θ}}\right)}{r}$

$\=\frac{w_{\mathrm{\θ} r}-v_z-r v_{z r}}{r}\+\frac{u_{z \mathrm{\θ}}-w_{r \mathrm{\θ}}}{r}\+\frac{v_z\+r v_{r z}-u_{\mathrm{\θ} z}}{r}$

$\=\frac{\left(w_{\mathrm{\θ} r}-w_{r \mathrm{\θ}}\right)}{r}\+\frac{v_z-v_z}{r}\+\left(v_{r z}-v_{z r}\right)\+\frac{u_{z \mathrm{\θ}}-u_{\mathrm{\θ} z}}{r}$

$\=0$

Initialize

Loading Student:-VectorCalculus

Define the cylindrical vector field F

$⟨u\left(r\,\mathrm{\θ}\,z\right)\,v\left(r\,\mathrm{\θ}\,z\right)\,w\left(r\,\mathrm{\θ}\,z\right)⟩$ = $\left[\begin{array}{c}u\left(r\,\mathrm{\theta }\,z\right)\\ v\left(r\,\mathrm{\theta }\,z\right)\\ w\left(r\,\mathrm{\theta }\,z\right)\end{array}\right]$$\stackrel{\text{apply coordinates}}{\to }$$\left[\begin{array}{c}u\left(r\,\mathrm{\theta }\,z\right)\\ v\left(r\,\mathrm{\theta }\,z\right)\\ w\left(r\,\mathrm{\theta }\,z\right)\end{array}\right]$$\stackrel{\text{to Vector Field}}{\to }$$\left[\begin{array}{c}u\left(r\,\mathrm{\theta }\,z\right)\\ v\left(r\,\mathrm{\theta }\,z\right)\\ w\left(r\,\mathrm{\theta }\,z\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

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

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

$\mathbf{F}≔\mathrm{VectorField}\left(⟨u\left(r\,\mathrm{\θ}\,z\right)\,v\left(r\,\mathrm{\θ}\,z\right)\,w\left(r\,\mathrm{\θ}\,z\right)⟩\,\mathrm{cylindrical}\left[r\,\mathrm{\θ}\,z\right]\right)$ = $$

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

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