Initialize

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

Install a notation-improving device

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,g\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,h\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,\mathrm{quiet}\right)$

Use the VectorField command to define F as a vector field in spherical coordinates

$\mathbf{F}≔\mathrm{VectorField}\left(⟨f\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,g\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)\,h\left(\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right)⟩\,\mathrm{spherical}\left[\mathrm{\ρ}\,\mathrm{\φ}\,\mathrm{\θ}\right]\right)$

$\left[\begin{array}{c}f\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\\ g\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\\ h\left(\mathrm{\rho }\,\mathrm{\phi }\,\mathrm{\theta }\right)\end{array}\right]$

Use the MapToBasis command to change to Cartesian coordinates

$\mathbf{G}≔\mathrm{MapToBasis}\left(\mathbf{F}\,\mathrm{cartesian}\left[x\,y\,z\right]\right)$

$\left[\begin{array}{c}\frac{f\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x}{\sqrt{x^2\+y^2\+z^2}}\+\frac{g\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)xz}{\sqrt{x^2\+y^2}\sqrt{x^2\+y^2\+z^2}}-\frac{h\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y}{\sqrt{x^2\+y^2}}\\ \frac{f\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y}{\sqrt{x^2\+y^2\+z^2}}\+\frac{g\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)yz}{\sqrt{x^2\+y^2}\sqrt{x^2\+y^2\+z^2}}\+\frac{h\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x}{\sqrt{x^2\+y^2}}\\ \frac{f\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)z}{\sqrt{x^2\+y^2\+z^2}}\+\frac{g\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\left(-x^2-y^2\right)}{\sqrt{x^2\+y^2}\sqrt{x^2\+y^2\+z^2}}\end{array}\right]$

Use the Curl command to obtain the curl of this Cartesian vector field

$\mathbf{curlG}≔\mathrm{simplify}\left(\mathrm{Curl}\left(\mathbf{G}\right)\right)$

$\left[\begin{array}{c}-\frac{{\mathrm{D}}_1\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y\sqrt{x^2\+y^2}{x}^2\+{\mathrm{D}}_1\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}{y}^3\+{\mathrm{D}}_1\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}y{z}^2\+{\mathrm{D}}_1\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\sqrt{x^2\+y^2\+z^2}\sqrt{x^2\+y^2}z-{\mathrm{D}}_2\left(f\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2\+z^2}\sqrt{x^2\+y^2}y-{\mathrm{D}}_3\left(f\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\sqrt{x^2\+y^2\+z^2}z\+{\mathrm{D}}_3\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\sqrt{x^2\+y^2\+z^2}\sqrt{x^2\+y^2}\+g\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y\sqrt{x^2\+y^2\+z^2}\sqrt{x^2\+y^2}-{\mathrm{D}}_2\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x^3-{\mathrm{D}}_2\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x{y}^2}{\left(x^2\+y^2\right)\left(x^2\+y^2\+z^2\right)}\\ \frac{{\mathrm{D}}_1\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}{x}^3\+{\mathrm{D}}_1\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\sqrt{x^2\+y^2}{y}^2\+{\mathrm{D}}_1\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\sqrt{x^2\+y^2}{z}^2-{\mathrm{D}}_1\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2\+z^2}\sqrt{x^2\+y^2}yz\+{\mathrm{D}}_3\left(f\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2\+z^2}yz-{\mathrm{D}}_3\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y\sqrt{x^2\+y^2\+z^2}\sqrt{x^2\+y^2}-{\mathrm{D}}_2\left(f\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\sqrt{x^2\+y^2\+z^2}\sqrt{x^2\+y^2}\+g\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\sqrt{x^2\+y^2\+z^2}\sqrt{x^2\+y^2}\+{\mathrm{D}}_2\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x^{2}y\+{\mathrm{D}}_2\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y^3}{\left(x^2\+y^2\right)\left(x^2\+y^2\+z^2\right)}\\ -\frac{-{\mathrm{D}}_1\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x^4-2{\mathrm{D}}_1\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x^2{y}^2-{\mathrm{D}}_1\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x^2{z}^2-h\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2\+z^2}{x}^2\+{\mathrm{D}}_3\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x^{2}z\+{\mathrm{D}}_3\left(f\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}{x}^2-{\mathrm{D}}_1\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y^4-{\mathrm{D}}_1\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y^2{z}^2-h\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2\+z^2}{y}^2-h\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2\+z^2}{z}^2\+{\mathrm{D}}_3\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y^{2}z\+{\mathrm{D}}_3\left(g\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)z^3\+{\mathrm{D}}_3\left(f\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}{y}^2\+{\mathrm{D}}_3\left(f\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}{z}^2-{\mathrm{D}}_2\left(h\right)\left(\sqrt{x^2\+y^2\+z^2}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}\sqrt{x^2\+y^2\+z^2}z}{\sqrt{x^2\+y^2}{\left(x^2\+y^2\+z^2\right)}^{3\/2}}\end{array}\right]$

$\mathbf{curlGs}≔\mathrm{simplify}\left(\mathrm{eval}\left(\mathbf{curlG}\,x^2\+y^2\+z^2\={\mathrm{\rho }}^2\right)\right) assuming \mathrm{\ρ}\>0$

$\left[\begin{array}{c}-\frac{{\mathrm{D}}_1\left(g\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y\sqrt{x^2\+y^2}{x}^2\+{\mathrm{D}}_1\left(g\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}{y}^3\+{\mathrm{D}}_1\left(g\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}y{z}^2\+{\mathrm{D}}_1\left(h\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\mathrm{\ρ}\sqrt{x^2\+y^2}z-{\mathrm{D}}_2\left(f\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\mathrm{\ρ}\sqrt{x^2\+y^2}y\+g\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y\mathrm{\ρ}\sqrt{x^2\+y^2}\+{\mathrm{D}}_3\left(g\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\mathrm{\ρ}\sqrt{x^2\+y^2}-{\mathrm{D}}_3\left(f\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\mathrm{\ρ}z-{\mathrm{D}}_2\left(h\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x^3-{\mathrm{D}}_2\left(h\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x{y}^2}{\left(x^2\+y^2\right){\mathrm{\ρ}}^2}\\ \frac{{\mathrm{D}}_1\left(g\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\sqrt{x^2\+y^2}{x}^3\+{\mathrm{D}}_1\left(g\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\sqrt{x^2\+y^2}{y}^2\+{\mathrm{D}}_1\left(g\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\sqrt{x^2\+y^2}{z}^2-{\mathrm{D}}_1\left(h\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\mathrm{\ρ}\sqrt{x^2\+y^2}yz\+{\mathrm{D}}_3\left(f\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)\mathrm{\ρ}yz-{\mathrm{D}}_3\left(g\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y\mathrm{\ρ}\sqrt{x^2\+y^2}-{\mathrm{D}}_2\left(f\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\mathrm{\ρ}\sqrt{x^2\+y^2}\+g\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x\mathrm{\ρ}\sqrt{x^2\+y^2}\+{\mathrm{D}}_2\left(h\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)x^{2}y\+{\mathrm{D}}_2\left(h\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,z\right)\,\arctan \left(y\,x\right)\right)y^3}{\left(x^2\+y^2\right){\mathrm{\ρ}}^2}\\ \frac{{\mathrm{D}}_1\left(h\right)\left(\mathrm{\ρ}\,\arctan \left(\sqrt{x^2\+y^2}\,\right)\right)}{}\end{array}\right]$