Recast a given vector field F in cylindrical coordinates

[Bestphysics112]

Homework Statement

F(x,y,z) = xzi

Homework Equations

N/A

The Attempt at a Solution

I just said that x = rcos(θ) so F(r,θ,z) = rcos(θ)z. Is this correct? Beaucse I am also asked to find curl of F in Cartesian coordinates and compare to curl of F in cylindrical coordinates. For Curl of F in cylindrical coordinates I obtained rcos(θ)e_θ+sin(θ)ze_z. This doesn't look anything like the curl found in Cartesian coordinates. Where am i going wrong?

 

[Charles Link]

The unit vector $\hat{i}$ also needs to be put in cylindrical coordinates. $\hat{i}=cos(\theta) \hat{a}_r -sin(\theta) \hat{a}_{\theta}$ if I computed it correctly. Once you do that, you should get agreement when you do the curl operation in cylindrical coordinates. $\\$ Editing: Yes, I computed it, and got agreement. I'd be happy to check the answer for you that you get.

 

[Bestphysics112]

The unit vector $\hat{i}$ also needs to be put in cylindrical coordinates. $\hat{i}=cos(\theta) \hat{a}_r -sin(\theta) \hat{a}_{\theta}$ if I computed it correctly. Once you do that, you should get agreement when you do the curl operation in cylindrical coordinates. $\\$ Editing: Yes, I computed it, and got agreement. I'd be happy to check the answer for you that you get.

Hello! Sorry I just saw this reply. As an answer, I got (rcos(θ)sin(θ)e_r +rcos²(θ)e_θ +(2cos(θ)sin(θ)z+ 2sin(θ)cos(θ)z)e_z

Is this answer correct?

Edit- Yes! I figured out my mistake and I got an equivalent answer. Thank you for the help Charles

 

[Charles Link]

Hello! Sorry I just saw this reply. As an answer, I got (rcos(θ)sin(θ)e_r +rcos²(θ)e_θ +(2cos(θ)sin(θ)z+ 2sin(θ)cos(θ)z)e_z

Is this answer correct?

Edit- Yes! I figured out my mistake and I got an equivalent answer. Thank you for the help Charles

Please check your $e_z$ term. The first term gets a minus sign so that the $e_z$ term is zero. Also what did you get in Cartesian coordinates for the curl? The two should agree and I think they do if the $e_z$ term is zero.

 

[Bestphysics112]

Please check your $e_z$ term. The first term gets a minus sign so that the $e_z$ term is zero. Also what did you get in Cartesian coordinates for the curl? The two should agree and I think they do if the $e_z$ term is zero.

I don't have my homework with me, but I forgot to edit my post with the answer I got. The e_z terms canceled