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Cylindrical Coordinate System. Please check my answer

  1. Jan 31, 2012 #1
    1. The problem statement, all variables and given/known data

    (a) In cylindrical coordinates , show that [itex]\hat{r}[/itex] points along the x-axis is [itex]\phi[/itex] = 0 .

    (b) In what direction is [itex]\hat{\phi}[/itex] if [itex]\phi[/itex] = 90°

    2. Relevant equations



    3. The attempt at a solution

    here is my solution. for a.

    [itex] \vec{r} = \rho cos \phi \hat{i} + \rho sin \phi \hat{j} + z \hat{k} [/itex]

    [itex]\frac{\partial \vec{r}}{\partial \rho}[/itex] = [itex]\frac{\partial \rho}{\partial {\rho}}[/itex] cos [itex]\phi[/itex] [itex]\hat{i}[/itex] + [itex]\frac{\partial \rho}{\partial {\rho}}[/itex] sin [itex]\phi[/itex] [itex]\hat{j}[/itex]

    [itex]\frac{\partial \vec{r}}{\partial \rho}[/itex] = [itex] cos \phi \hat{i} + sin \phi \hat{j}[/itex]

    [itex]\left|\frac{\partial \vec{r}}{\partial \rho}\right|[/itex] = [itex]\sqrt{cos^2 \phi + sin^2 \phi}[/itex]


    [itex]\left|\frac{\partial \vec{r}}{\partial \rho}\right|[/itex] = 1

    [itex] \hat{r} = \frac{\frac{\partial \vec{r}}{\partial \rho}}{\left|\frac{\partial \vec{r}}{\partial \rho}\right|} [/itex]


    [itex] \hat{r} [/itex] = [itex] cos \phi \hat{i} + sin \phi \hat{j}[/itex]

    so if [itex] \phi [/itex] is 0.

    [itex] \hat{r} = \hat{i} [/itex]

    meaning [itex] \hat{r} [/itex] is pointing at the direction of the positive x-axis


    now for b.

    [itex] \vec{r} = \rho cos \phi \hat{i} + \rho sin \phi \hat{j} + z \hat{k} [/itex]

    [itex]\frac{\partial \vec{r}}{\partial \phi}[/itex] = [itex] \rho \frac{\partial cos \phi}{\partial {\phi}}[/itex] [itex]\hat{i}[/itex] + [itex] \rho \frac{\partial sin \phi}{\partial {\phi}}[/itex]

    [itex]\frac{\partial \vec{r}}{\partial \phi}[/itex] = [itex] - \rho sin \phi \hat{i} + \rho cos \phi \hat{j} [/itex]



    [itex]\left|\frac{\partial \vec{r}}{\partial \phi}\right|[/itex] = [itex]\sqrt{\rho^2 (sin^2 \phi + cos^2 \phi)}[/itex]


    [itex]\left|\frac{\partial \vec{r}}{\partial \phi}\right| = \rho [/itex]


    [itex] \hat{\phi} = \frac{\frac{\partial \vec{r}}{\partial \phi}}{\left|\frac{\partial \vec{r}}{\partial \phi}\right|} [/itex]

    [itex] \hat{\phi} = -sin \phi \hat{i} + cos \phi \hat{j} [/itex]

    so if phi is 90°

    [itex] \hat{\phi} = -\hat{i} [/itex]

    meaning [itex] \hat{\phi} [/itex] points along the negative x-axis


    I hope I'm correct. can someone please tell me if I did this right? Thanks
     
    Last edited: Jan 31, 2012
  2. jcsd
  3. Jan 31, 2012 #2
    You changed notation in b when the norm of the partial was taken (it should be the partial w.r.t [itex]\phi[/itex]), but other than that it is all correct.
     
  4. Jan 31, 2012 #3
    ow, I'm sorry about that. I just copy paste it from letter a's equation. wasn't able to change it. sorry. Thanks for the reminder though. :))
     
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