Cylindrical Coordinate System. Please check my answer

[jhosamelly]

Homework Statement

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

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

Homework Equations

The Attempt at a Solution

here is my solution. for a.

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

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

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

\left|\frac{\partial \vec{r}}{\partial \rho}\right| = \sqrt{cos^2 \phi + sin^2 \phi}\left|\frac{\partial \vec{r}}{\partial \rho}\right| = 1

\hat{r} = \frac{\frac{\partial \vec{r}}{\partial \rho}}{\left|\frac{\partial \vec{r}}{\partial \rho}\right|}\hat{r} = cos \phi \hat{i} + sin \phi \hat{j}

so if \phi is 0.

\hat{r} = \hat{i}

meaning \hat{r} is pointing at the direction of the positive x-axisnow for b.

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

\frac{\partial \vec{r}}{\partial \phi} = \rho \frac{\partial cos \phi}{\partial {\phi}} \hat{i} + \rho \frac{\partial sin \phi}{\partial {\phi}}

\frac{\partial \vec{r}}{\partial \phi} = - \rho sin \phi \hat{i} + \rho cos \phi \hat{j}
\left|\frac{\partial \vec{r}}{\partial \phi}\right| = \sqrt{\rho^2 (sin^2 \phi + cos^2 \phi)}\left|\frac{\partial \vec{r}}{\partial \phi}\right| = \rho \hat{\phi} = \frac{\frac{\partial \vec{r}}{\partial \phi}}{\left|\frac{\partial \vec{r}}{\partial \phi}\right|}

\hat{\phi} = -sin \phi \hat{i} + cos \phi \hat{j}

so if phi is 90°

\hat{\phi} = -\hat{i}

meaning \hat{\phi} points along the negative x-axisI hope I'm correct. can someone please tell me if I did this right? Thanks

 

[δοτ]

You changed notation in b when the norm of the partial was taken (it should be the partial w.r.t \phi), but other than that it is all correct.

 

[jhosamelly]

You changed notation in b when the norm of the partial was taken (it should be ht epartial w.r.t φ), but other than that it is all correct.

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.