derivatives in cylindrical coord's, need help

*To*: mathgroup at smc.vnet.net*Subject*: [mg51268] derivatives in cylindrical coord's, need help*From*: "symbio" <symbio at h0tmail.com>*Date*: Mon, 11 Oct 2004 01:25:27 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Hello, 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 CYLINDRICAL coord's. 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.