Cylindrical Conversion

{Sqrt[x^2+y^2],ArcTan[y/x],z}/.{x->3,y->-3,z->7}

{r Cos[theta],r Sin[theta],z}/.{r->2,theta->2Pi/3,z->7}

FullSimplify[x^2+y^2/.{x->r Cos[theta],y->r Sin[theta]}]

Cylindrical Plotting

Code from https://mathematica.stackexchange.com/a/19084

CylindricalPlot3D[f_,{rMin_,rMax_},{tMin_,tMax_},opts___]:=ParametricPlot3D[{r Cos[t],r Sin[t],f[r,t]},{r,rMin,rMax},{t,tMin,tMax},opts]

f[r_,t_]:=r^2 Sin[3t];
CylindricalPlot3D[f,{0,1},{0,2Pi},BoxRatios->{1,1,1}]

f[r_,t_]:=r^2 ;
CylindricalPlot3D[f,{0,1},{0,2Pi},BoxRatios->{1,1,1}]

Spherical Conversion

{ρ->Sqrt[x^2+y^2+z^2],
θ->ArcTan[x,y],
ϕ->ArcTan[Sqrt[x^2+y^2]/z]
}/.{x->-1,y->1,z->Sqrt[6]}

{x->ρ Sin[ϕ]Cos[θ],
y->ρ Sin[ϕ]Sin[θ],
z->ρ Cos[ϕ]
}/.{ρ->2,θ->Pi/4,ϕ->Pi/3}

FullSimplify[rho*Sin[phi]/.{rho->Sqrt[x^2+y^2+z^2],theta->ArcTan[x,y],phi->ArcTan[Sqrt[x^2+y^2]/z]}]

FullSimplify[z-x^2+y^2/.{x->ρ Sin[ϕ]Cos[θ],y->ρ Sin[ϕ]Sin[θ],z->ρ Cos[ϕ]}]

Solve[(Cos[ϕ]-ρ Cos[2 θ] ((Sin[ϕ])^2))==0,ρ]

SphericalPlot3D[Cot[ϕ] Csc[ϕ] Sec[2 θ],{ϕ,0,Pi},{θ,-Pi/4,Pi/4},PlotPoints->200,PlotRange->10]

Spherical Plotting

Mathematica’s conventions for theta and phi are reversed.

rho*Sin[phi]=2 -> rho = 2/Sin[phi]

SphericalPlot3D[2/Sin[phi],{phi,0,Pi},{theta,0,2Pi},BoxRatios->{1,1,1}]

SphericalPlot3D[2Cos[phi],{phi,0,Pi},{theta,0,2Pi},BoxRatios->{1,1,1}]

SphericalPlot3D[Sin[4theta]Cos[3phi],{phi,0,Pi},{theta,0,2Pi},PlotRange->1,BoxRatios->{1,1,1}]

SphericalPlot3D[Sin[4theta]Cos[3phi],{phi,Pi/6,3Pi/6},{theta,0,Pi/4},PlotRange->1,BoxRatios->{1,1,1}]