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derivatives in cylindrical coord's, need help

I'm trying to do a EM problem from beginning to end, all in cylindrical
coordinates, without converting to Cartesian at all.  The problem I'm trying
to solve is quite simple, it is:  Calculate the flux through a surface given
the B-field (magnetic flux density), and the surface of a circle sitting in
x-y plane at z=0, having a radius R.   The B-field is given to be= {0, 0, 1
/ (1 + rho) }, where rho is radial distance in cylindrical coord's, and the
B-field is oriented in the unit vector direction z ( where z is the unit
vector in z direction from cylindrical coord's{rho, phi, z} ).  So, this is
just a double vector integral of a field through a surface:  Integral [ B .
ds].  The way I do it is as follows:

0.  Given B-feild as ==> B[rho, phi, z] = {0, 0, 1 / (1 + rho) }, in
1.  Define surface parameterization for circle, in CYLINDRICAL coord's, ==>
path[rho, phi] = {rho, phi, 0}
2.  Take derivative D/D'rho' path[rho, phi]  and assign to vector v1 = {1,
0, 0}, in CYLINDRICAL coord's.
3.  Take derivative D/D'phi' path[rho, phi] and assign to vector v2 = {0, 1,
0}, in CYLINDRICAL coord's.
4.  Derive normal vector to surface by taking Cross Product of v1 and v2
above, in CYLINDRICAL coord's, ==> normal = {0, 0, 0} !!!!!  Step#4 is where
I get stuck, and cannot move forward!!
5.  The next step would be have been to take the Dot Product B [ path [ rho,
phi] ] with normal vector above, in CYLINDRICAL coord's, but normal is not
correct and so this step will not work
6.  The final step would have been to calculate the integrand above (result
of dot product in cylindrical coord's) in cylindrical coord's for {phi, 0,
2*Pi}, and then {rho, 0, R}, but again since normal vector is 0,0,0, this we
are stuck on step #4 above.

So, my question is what am I doing wrong in step 4 first?  I guess my
derivative in cylindrical coord's is wrong, please help.

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