Question

Express the cylindrical unit vectors $\hat{\mathbf{s}}\mathbf{,}\mathbf{}\hat{\mathbf{\varphi }}\mathbf{,}\mathbf{}\hat{\mathbf{z}}$ in terms of $\hat{\mathbf{x}}\mathbf{,}\mathbf{}\hat{\mathbf{y}}\mathbf{,}\mathbf{}\hat{\mathbf{z}}$ (that is, derive Eq. 1.75). "Invert" your formulas to get $\hat{\mathbf{x}}\mathbf{,}\mathbf{}\hat{\mathbf{y}}\mathbf{,}\mathbf{}\hat{\mathbf{z}}$in terms of $\hat{\mathbf{s}}\mathbf{,}\mathbf{}\hat{\mathbf{\varphi }}\mathbf{,}\mathbf{}\hat{\mathbf{z}}$

Short answer

It is obtained that$\overbrace{x}=\left(\cos \psi \right)\stackrel{⏜}{s}-\left(\sin \psi \right)\stackrel{⏜}{\varphi },\stackrel{⏜}{y}=\left(\sin \psi \right)\stackrel{⏜}{s}+\left(\cos \psi \right)\stackrel{⏜}{s},and\stackrel{⏜}{z}=\stackrel{⏜}{z}.$

Step-by-step solution

Define cylindrical coordinates

In cylindrical coordinates, the point is represented as $P=(s,\varphi ,z)$ , where $P=(s,\varphi ,z)$ 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:

$$\begin{gathered} x=s\cos \varphi \\ y=s\sin \varphi \\ z=z \end{gathered}$$

The unit vectors in cylindrical coordinates are:

$$\begin{gathered} \stackrel{⏜}{s}=\left(\cos \varphi \right)\stackrel{⏜}{x}+\left(\sin \varphi \right)\stackrel{⏜}{y} \\ \stackrel{⏜}{\varphi }=-\left(\sin \varphi \right)\stackrel{⏜}{x}+\left(\cos \varphi \right)\stackrel{⏜}{y} \\ \stackrel{⏜}{z}=\stackrel{⏜}{z} \end{gathered}$$

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

$$\begin{gathered} dx=\left(\cos \varphi \right)ds \\ dy=\left(\sin \varphi \right)ds \\ dz=0 \\ \mathrm{The}\mathrm{displacement}\mathrm{vector}\mathrm{now}\mathrm{becomes}: \\ \mathrm{dl}=\left(\mathrm{cos\varphi }\right)\mathrm{ds}\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi }\right)\mathrm{ds}\stackrel{⏜}{\mathrm{y}}+0\stackrel{⏜}{\mathrm{z}} \\ =\mathrm{ds}\left(\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}}\right) \\ \mathrm{Compare}\mathrm{above}\mathrm{equation}\mathrm{with}\mathrm{dl}=\mathrm{ds}\stackrel{⏜}{\mathrm{s}},\mathrm{we}\mathrm{get} \\ \stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{y}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}} \end{gathered}$$

Step: 2 Compute unit vector s^ .

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

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

The displacement vector now becomes:

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

Compare above equation with $dl=sd\varphi \hat{\varphi }$ , we get,

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

Step: 3 Compute unit vector  x^

$$\begin{gathered} As\stackrel{⏜}{s}=\left(\cos \varphi \right)\stackrel{⏜}{x}+\left(\sin \varphi \right)\stackrel{⏜}{y},\mathrm{multiple}\mathrm{cos\phi }\mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{of} \\ \stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}}\mathrm{as}, \\ \left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}=\left({\cos }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi cos\varphi }\right)\stackrel{⏜}{\mathrm{y}} \\ \mathrm{Now},\mathrm{multiply}\sin \varphi \mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{of}\stackrel{⏜}{\varphi }=-\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{y}}\mathrm{as}, \\ \left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{\varphi }}=\left(-{\sin }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\sin \varphi c\mathrm{os}\varphi \right)\stackrel{⏜}{\mathrm{y}} \end{gathered}$$

$$\begin{gathered} \mathrm{subtract}\left(\sin \phi \right)\stackrel{⏜}{\varphi }=\left(-{\sin }^2\varphi \right)\stackrel{⏜}{x}+\left(\sin \varphi \cos \varphi \right)\stackrel{⏜}{y}\mathrm{from} \\ \left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi cos\varphi }\right)\stackrel{⏜}{\mathrm{y}}\mathrm{as}, \\ \left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}=-\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{\varphi }}=\left({\cos }^2\mathrm{\varphi }+{\sin }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{x}} \\ \stackrel{⏜}{\mathrm{x}}=\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}-\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{\varphi }} \end{gathered}$$

Step: 4 Compute unit vector y^ 

$$\begin{gathered} \mathrm{As}\stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}},\mathrm{multiply}\sin \varphi \mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{of} \\ \stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}}\mathrm{as}, \\ \left(\sin \varphi \right)\stackrel{⏜}{\mathrm{x}}=\left(\sin \varphi \cos \varphi \right)\stackrel{⏜}{\mathrm{x}}+\left({\sin }^2\varphi \right)\stackrel{⏜}{\mathrm{y}} \\ \mathrm{Now},\mathrm{multiply}\cos \varphi \mathrm{on}\mathrm{both}\mathrm{sides}\mathrm{of}\stackrel{⏜}{\mathrm{\varphi }}=\left(-\sin \varphi \right)\stackrel{⏜}{\mathrm{x}}+\left(\cos \varphi \right)\stackrel{⏜}{\mathrm{y}}\mathrm{as}, \\ \left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{\varphi }}=-\left(-sin\varphi cos\varphi \right)\stackrel{⏜}{\mathrm{x}}+\left(\cos {}^2\varphi \right)\stackrel{⏜}{\mathrm{y}} \\ \mathrm{Add}\mathrm{equations}\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{\varphi }}=\left(-\mathrm{sin\varphi cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\cos {}^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{y}}\mathrm{and} \\ \left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{x}}=\left(\mathrm{sin\varphi cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left({\sin }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{y}}\mathrm{as}, \\ \left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{\varphi }}=\left({\cos }^2\mathrm{\varphi }+{\sin }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{y}} \\ \stackrel{⏜}{\mathrm{y}}=\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{s}}+\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{\varphi }} \\ \mathrm{Therefore},\mathrm{the}\mathrm{required}\mathrm{equations}\mathrm{are}\stackrel{⏜}{\mathrm{x}}=\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}-\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{x}}; \\ \stackrel{⏜}{\mathrm{y}}=\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{s}}+\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{\varphi }}\mathrm{and}\stackrel{⏜}{\mathrm{z}}=\stackrel{⏜}{\mathrm{z}}. \end{gathered}$$