91Ó°ÊÓ

In cylindrical coordinates, the point is represented as $P=(s,\mathrm{Ï•},z)$ , where $P=(s,\mathrm{Ï•},z)$ is distance of point P from z axis, the azimuthal angle, and coordinate of point P on z-axis respectively, as shown in following figure:

From the figure, write:

$$\begin{gathered} x=s\cos \mathrm{Ï•} \\ y=s\sin \mathrm{Ï•} \\ z=z \end{gathered}$$

The unit vectors in cylindrical coordinates are:

$$\begin{gathered} \stackrel{Áåœ}{s}=\left(\cos \mathrm{Ï•}\right)\stackrel{Áåœ}{x}+\left(\sin \mathrm{Ï•}\right)\stackrel{Áåœ}{y} \\ \stackrel{Áåœ}{\mathrm{Ï•}}=-\left(\sin \mathrm{Ï•}\right)\stackrel{Áåœ}{x}+\left(\cos \mathrm{Ï•}\right)\stackrel{Áåœ}{y} \\ \stackrel{Áåœ}{z}=\stackrel{Áåœ}{z} \end{gathered}$$

The displacement vector is given as $dl=dx\hat{x}+dy\hat{y}+dz\hat{z}.$Differentiate transformation equation with respect to s .

$$\begin{gathered} dx=\left(\cos \mathrm{Ï•}\right)ds \\ dy=\left(\sin \mathrm{Ï•}\right)ds \\ dz=0 \\ \mathrm{The}\mathrm{displacement}\mathrm{vector}\mathrm{now}\mathrm{becomes}: \\ \mathrm{dl}=\left(\mathrm{³¦´Ç²õÏ•}\right)\mathrm{ds}\stackrel{Áåœ}{\mathrm{x}}+\left(\mathrm{²õ¾\pm ²ÔÏ•}\right)\mathrm{ds}\stackrel{Áåœ}{\mathrm{y}}+0\stackrel{Áåœ}{\mathrm{z}} \\ =\mathrm{ds}\left(\left(\mathrm{³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\mathrm{²õ¾\pm ²ÔÏ•}\right)\stackrel{Áåœ}{\mathrm{y}}\right) \\ \mathrm{Compare}\mathrm{above}\mathrm{equation}\mathrm{with}\mathrm{dl}=\mathrm{ds}\stackrel{Áåœ}{\mathrm{s}},\mathrm{we}\mathrm{get} \\ \stackrel{Áåœ}{\mathrm{s}}=\left(\mathrm{³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{y}}+\left(\mathrm{²õ¾\pm ²ÔÏ•}\right)\stackrel{Áåœ}{\mathrm{y}} \end{gathered}$$

The displacement vector is given as$d{l}_{\mathrm{Ï•}}=dx\hat{x},dy\hat{y},dz\hat{z}$ . Differentiate transformation equation with respect to $\mathrm{Ï•}$.

$$\begin{gathered} dx=s\sin \mathrm{Ï•}d\mathrm{Ï•} \\ dy=s\cos \mathrm{Ï•}d\mathrm{Ï•} \\ dz=0 \end{gathered}$$

The displacement vector now becomes:

$$\begin{gathered} dl=s\sin \mathrm{Ï•}d\mathrm{Ï•}\hat{x}+s\cos \mathrm{Ï•}d\mathrm{Ï•}\hat{y}+0\hat{z} \\ =ds\left((s\sin \mathrm{Ï•}d\mathrm{Ï•})\hat{x}+\left(s\cos \mathrm{Ï•}d\mathrm{Ï•}\right)\hat{y}\right) \end{gathered}$$

Compare above equation with $dl=sd\mathrm{Ï•}\widehat{\mathrm{Ï•}}$ , we get,

$\hat{s}=(-\sin \mathrm{Ï•})\hat{x}+\left(\cos \mathrm{Ï•}\right)\hat{y}$

$$\begin{gathered} As\stackrel{Áåœ}{s}=\left(\cos \mathrm{Ï•}\right)\stackrel{Áåœ}{x}+\left(\sin \mathrm{Ï•}\right)\stackrel{Áåœ}{y},\mathrm{multiple}\mathrm{³¦´Ç²õφ}\mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{of} \\ \stackrel{Áåœ}{\mathrm{s}}=\left(\mathrm{³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\mathrm{²õ¾\pm ²ÔÏ•}\right)\stackrel{Áåœ}{\mathrm{y}}\mathrm{as}, \\ \left(\mathrm{³¦´Ç²õφ}\right)\stackrel{Áåœ}{\mathrm{s}}=\left({\cos }^2\mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\mathrm{²õ¾\pm ²ÔÏ•³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{y}} \\ \mathrm{Now},\mathrm{multiply}\sin \mathrm{Ï•}\mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{of}\stackrel{Áåœ}{\mathrm{Ï•}}=-\left(\mathrm{²õ¾\pm ²ÔÏ•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\mathrm{³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{y}}\mathrm{as}, \\ \left(\mathrm{²õ¾\pm ²Ôφ}\right)\stackrel{Áåœ}{\mathrm{Ï•}}=\left(-{\sin }^2\mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\sin \mathrm{Ï•}c\mathrm{os}\mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{y}} \end{gathered}$$

$$\begin{gathered} \mathrm{As}\stackrel{Áåœ}{\mathrm{s}}=\left(\mathrm{³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\mathrm{²õ¾\pm ²ÔÏ•}\right)\stackrel{Áåœ}{\mathrm{y}},\mathrm{multiply}\sin \mathrm{Ï•}\mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{of} \\ \stackrel{Áåœ}{\mathrm{s}}=\left(\mathrm{³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\mathrm{²õ¾\pm ²ÔÏ•}\right)\stackrel{Áåœ}{\mathrm{y}}\mathrm{as}, \\ \left(\sin \mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{x}}=\left(\sin \mathrm{Ï•}\cos \mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left({\sin }^2\mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{y}} \\ \mathrm{Now},\mathrm{multiply}\cos \mathrm{Ï•}\mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{of}\stackrel{Áåœ}{\mathrm{Ï•}}=\left(-\sin \mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\cos \mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{y}}\mathrm{as}, \\ \left(\mathrm{³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{Ï•}}=-\left(-sin\mathrm{Ï•}cos\mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\cos {}^2\mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{y}} \\ \mathrm{Add}\mathrm{equations}\left(\mathrm{³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{Ï•}}=\left(-\mathrm{²õ¾\pm ²ÔÏ•³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\cos {}^2\mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{y}}\mathrm{and} \\ \left(\mathrm{²õ¾\pm ²ÔÏ•}\right)\stackrel{Áåœ}{\mathrm{x}}=\left(\mathrm{²õ¾\pm ²ÔÏ•³¦´Ç²õÏ•}\right)\stackrel{Áåœ}{\mathrm{x}}+\left({\sin }^2\mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{y}}\mathrm{as}, \\ \left(\mathrm{²õ¾\pm ²Ôφ}\right)\stackrel{Áåœ}{\mathrm{x}}+\left(\mathrm{³¦´Ç²õφ}\right)\stackrel{Áåœ}{\mathrm{Ï•}}=\left({\cos }^2\mathrm{Ï•}+{\sin }^2\mathrm{Ï•}\right)\stackrel{Áåœ}{\mathrm{y}} \\ \stackrel{Áåœ}{\mathrm{y}}=\left(\mathrm{²õ¾\pm ²Ôφ}\right)\stackrel{Áåœ}{\mathrm{s}}+\left(\mathrm{³¦´Ç²õφ}\right)\stackrel{Áåœ}{\mathrm{Ï•}} \\ \mathrm{Therefore},\mathrm{the}\mathrm{required}\mathrm{equations}\mathrm{are}\stackrel{Áåœ}{\mathrm{x}}=\left(\mathrm{³¦´Ç²õφ}\right)\stackrel{Áåœ}{\mathrm{s}}-\left(\mathrm{²õ¾\pm ²Ôφ}\right)\stackrel{Áåœ}{\mathrm{x}}; \\ \stackrel{Áåœ}{\mathrm{y}}=\left(\mathrm{²õ¾\pm ²Ôφ}\right)\stackrel{Áåœ}{\mathrm{s}}+\left(\mathrm{³¦´Ç²õφ}\right)\stackrel{Áåœ}{\mathrm{Ï•}}\mathrm{and}\stackrel{Áåœ}{\mathrm{z}}=\stackrel{Áåœ}{\mathrm{z}}. \end{gathered}$$