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cross-product in cylindrical problem


I'm really puzzled by this behavior of Mathematica, I have two vectors in 
cylindrical coordinates and would like to take their cross-product in 
cylindrical, but it seems to give me incorrect answer, see below:

define parametric path {r,phi,z}

In[110]:=
f[\[Rho]_, \[Phi]_] = {\[Rho], \[Phi], 0}
Out[110]=
{\[Rho], \[Phi], 0}

take derivates of path w.r.t. r then w.r.t phi, get {1,0,0}, and {0,1,0}

In[113]:=
v1 = D[f[\[Rho], \[Phi]], \[Rho]]
v2 = D[f[\[Rho], \[Phi]], \[Phi]]
Out[113]=
{1, 0, 0}
Out[114]=
{0, 1, 0}

then cross them in cylindrical coords, and should get {0,0,1}, but instead 
get wrong answer below

In[117]:=
n = CrossProduct[v1, v2, Cylindrical[\[Rho], \[Phi], z]] // FullSimplify
Out[117]=
{0, 0, 0}

As you can see, when I cross {1,0,0} with {0,1,0} in cylindrical coords, I 
get {0,0,0}, when I should be getting {0,0,1}.

Can anyone help?


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