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jhosamelly
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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
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