Plotting vector field in cylindrical co-ordinates

Question

I am trying to plot the following. Let $\Gamma^\top \Gamma=1$, and $e_3=\{0,0,1\}$ (unit vector along Z-axis). I am trying to plot $\dot \Gamma=e_3\times \Gamma$.

Tried the following: $\Gamma =\{\cos (\theta ) \cos (\phi ),\cos (\theta ) \sin (\phi ),\sin (\theta )\};$ and used $\text{VectorPlot}[\text{e3}\times \gamma ,\{\theta ,-\pi ,\pi \},\{\phi ,-\pi ,\pi \}]$ But VectorPlot says it need three argument.

Answer 1

Mathematica thinks (correctly) that your vector e3~Cross~\[CapitalGamma] is three-dimensional, and so doesn't know how to find a two-dimensional vector field to plot (which is what VectorPlot does).

If, as I assume, you want to plot the $x$- and $y$-components of your resultant field, you could do

\[CapitalGamma] = {Cos[\[Theta]] Cos[\[Phi]],Cos[\[Theta]] Sin[\[Phi]],Sin[\[Theta]]};
e3 = {0,0,1};
VectorPlot[(e3~Cross~\[CapitalGamma])[[1;;2]],{\[Theta],0,\[Pi]},{\[Phi],-\[Pi],\[Pi]}]

but you probably also want to convert $(\theta,\phi)$ to $(x,y)$.