Chapter 2, Problem 22P

To determine

Complete the given Cartesian coordinates expressions from the cylindrical and spherical coordinates.

Answer to Problem 22P

The completed Cartesian coordinates expressions of $\left[\begin{array}{c}{A}_x\\ A_y\\ A_z\end{array}\right]$ from the cylindrical and spherical coordinates are $\underset{\_}{\left[\begin{array}{ccc}\frac{x}{\sqrt{x^2+y^2}}& \frac{y}{\sqrt{x^2+y^2}}& 0\\ -\frac{y}{\sqrt{x^2+y^2}}& \frac{x}{\sqrt{x^2+y^2}}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{A}_{\rho }\\ A_{\varphi }\\ A_z\end{array}\right]}$ and

$\underset{\_}{\left[\begin{array}{ccc}\frac{x}{\sqrt{x^2+y^2+z^2}}& \frac{xz}{\sqrt{x^2+y^2+z^2}\sqrt{x^2+y^2}}& -\frac{y}{\sqrt{x^2+y^2}}\\ \frac{y}{\sqrt{x^2+y^2+z^2}}& \frac{yz}{\sqrt{x^2+y^2+z^2}\sqrt{x^2+y^2}}& \frac{x}{\sqrt{x^2+y^2}}\\ \frac{z}{\sqrt{x^2+y^2+z^2}}& -\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}& 0\end{array}\right]\left[\begin{array}{c}{A}_r\\ A_{\theta }\\ A_{\varphi }\end{array}\right]}$ respectively.

Explanation of Solution

Calculation:

Write the matrix form of expression to find to the Cartesian coordinates from cylindrical coordinates.

$\left[\begin{array}{c}{A}_x\\ A_y\\ A_z\end{array}\right]=\left[\begin{array}{ccc}\cos \varphi & -\sin \varphi & 0\\ \sin \varphi & \cos \varphi & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{A}_{\rho }\\ A_{\varphi }\\ A_z\end{array}\right]$        (1)

Consider the expressions for $\cos \varphi$ and $\sin \varphi$ in Cartesian coordinates.

$\cos \varphi =\frac{x}{\sqrt{x^2+y^2}}$        (2)

$\sin \varphi =\frac{y}{\sqrt{x^2+y^2}}$        (3)

Substitute Equations (2) and (3) in (1).

$\left[\begin{array}{c}{A}_x\\ A_y\\ A_z\end{array}\right]=\left[\begin{array}{ccc}\frac{x}{\sqrt{x^2+y^2}}& -\frac{y}{\sqrt{x^2+y^2}}& 0\\ \frac{y}{\sqrt{x^2+y^2}}& \frac{x}{\sqrt{x^2+y^2}}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{A}_{\rho }\\ A_{\varphi }\\ A_z\end{array}\right]$

Write the matrix form of expression to find to the Cartesian coordinates from spherical coordinates.

$\left[\begin{array}{c}{A}_x\\ A_y\\ A_z\end{array}\right]=\left[\begin{array}{ccc}\sin \theta \cos \varphi & \cos \theta \cos \varphi & -\sin \varphi \\ \sin \theta \sin \varphi & \cos \theta \sin \varphi & \cos \varphi \\ \cos \theta & -\sin \theta & 0\end{array}\right]\left[\begin{array}{c}{A}_r\\ A_{\theta }\\ A_{\varphi }\end{array}\right]$        (4)

Consider the expressions for $\sin \theta \cos \varphi$, $\sin \theta \sin \varphi$, $\cos \theta$ , and $\sin \theta$ in Cartesian coordinates form.

$\sin \theta \cos \varphi =\frac{x}{\sqrt{x^2+y^2+z^2}}$        (5)

$\sin \theta \sin \varphi =\frac{y}{\sqrt{x^2+y^2+z^2}}$        (6)

$\cos \theta =\frac{z}{\sqrt{x^2+y^2+z^2}}$        (7)

$\sin \theta =\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}$        (8)

Substitute Equations (2), (3, (5), (6), (7), and (8) in (4).

$\begin{array}{c}\left[\begin{array}{c}{A}_x\\ A_y\\ A_z\end{array}\right]=\left[\begin{array}{ccc}\frac{x}{\sqrt{x^2+y^2+z^2}}& \frac{z}{\sqrt{x^2+y^2+z^2}}\frac{x}{\sqrt{x^2+y^2}}& -\frac{y}{\sqrt{x^2+y^2}}\\ \frac{y}{\sqrt{x^2+y^2+z^2}}& \frac{z}{\sqrt{x^2+y^2+z^2}}\frac{y}{\sqrt{x^2+y^2}}& \frac{x}{\sqrt{x^2+y^2}}\\ \frac{z}{\sqrt{x^2+y^2+z^2}}& -\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}& 0\end{array}\right]\left[\begin{array}{c}{A}_r\\ A_{\theta }\\ A_{\varphi }\end{array}\right]\\ =\left[\begin{array}{ccc}\frac{x}{\sqrt{x^2+y^2+z^2}}& \frac{xz}{\sqrt{x^2+y^2+z^2}\sqrt{x^2+y^2}}& -\frac{y}{\sqrt{x^2+y^2}}\\ \frac{y}{\sqrt{x^2+y^2+z^2}}& \frac{yz}{\sqrt{x^2+y^2+z^2}\sqrt{x^2+y^2}}& \frac{x}{\sqrt{x^2+y^2}}\\ \frac{z}{\sqrt{x^2+y^2+z^2}}& -\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}& 0\end{array}\right]\left[\begin{array}{c}{A}_r\\ A_{\theta }\\ A_{\varphi }\end{array}\right]\end{array}$

Conclusion:

Thus, the completed Cartesian coordinates expressions of $\left[\begin{array}{c}{A}_x\\ A_y\\ A_z\end{array}\right]$ from the cylindrical and spherical coordinates are $\underset{\_}{\left[\begin{array}{ccc}\frac{x}{\sqrt{x^2+y^2}}& \frac{y}{\sqrt{x^2+y^2}}& 0\\ -\frac{y}{\sqrt{x^2+y^2}}& \frac{x}{\sqrt{x^2+y^2}}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{A}_{\rho }\\ A_{\varphi }\\ A_z\end{array}\right]}$ and

$\underset{\_}{\left[\begin{array}{ccc}\frac{x}{\sqrt{x^2+y^2+z^2}}& \frac{xz}{\sqrt{x^2+y^2+z^2}\sqrt{x^2+y^2}}& -\frac{y}{\sqrt{x^2+y^2}}\\ \frac{y}{\sqrt{x^2+y^2+z^2}}& \frac{yz}{\sqrt{x^2+y^2+z^2}\sqrt{x^2+y^2}}& \frac{x}{\sqrt{x^2+y^2}}\\ \frac{z}{\sqrt{x^2+y^2+z^2}}& -\frac{\sqrt{x^2+y^2}}{\sqrt{x^2+y^2+z^2}}& 0\end{array}\right]\left[\begin{array}{c}{A}_r\\ A_{\theta }\\ A_{\varphi }\end{array}\right]}$ respectively.