Express the plane $z=x$ in cylindrical and spherical coordinates.

This question aims to find the cylindrical and spherical coordinates of the plane $z = x$.

This question is based on the concept of coordinate systems from calculus. Cylindrical and spherical coordinate systems are expressed in the cartesian coordinate systems. A spherical object like a sphere of a ball is best expressed in a spherical coordinate system while cylindrical objects like pipes are best described in the cylindrical coordinate system.

The plane $z =x$ is a plane that lies in the $xz-plane$ in the cartesian coordinate system. The graph of plane $z=x$ is shown in Figure 1 and it can be seen that the $y$-component of the graph is zero.

We can express this plane in spherical and cylindrical coordinates using their derived formulas.

1) Cylindrical Coordinates are given by:

$(x, y, z) = (r \cos \theta, r \sin \theta, z) \quad 0 \leq \theta \leq 2\pi$

Where,

$r = \sqrt{x^2 + y^2} \quad r \geq 0$

Given,

$z = x$

So the equation becomes,

$(x, y, z) = (r \cos \theta, r \sin \theta, r \cos \theta)$

2) Spherical Coordinates are given by:

$(x, y, z) = (\rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi) \quad \rho \geq 0, 0 \leq \theta \leq 2\pi, 0 \leq \phi \leq \pi$

Given,

$z = x$

$\rho \cos \phi = \rho \sin \phi \cos \theta$

$\dfrac{\cos \phi}{\sin \phi} = \cos \theta$

$\cot \phi = \cos \theta$

$\theta = \arccos (\cot \phi)$

By substituting the values we get,

$(x, y, z) = (\rho \sin \phi \cos (\arccos (\cot \phi)), \rho \sin \phi \sin (\arccos (\cot \phi)), \rho \cos \phi)$

Simplifying by using trigonometric identities, we get:

$(x, y, z) = (\rho \cos \phi, \rho \sin \phi \sqrt{1 – \cot^{2} \phi}, \rho \cos \phi)$

Numerical Results:

Cylindrical Coordinates,

$(x, y, z) = (r \cos \theta, r \sin \theta, r \cos \theta)$

Spherical Coordinates,

$(x, y, z) = (\rho \cos \phi, \rho \sin \phi \sqrt{1 – \cot^{2} \phi}, \rho \cos \phi)$

Example:

Convert $(5, 2, 3)$ cartesian coordinates into cylindrical and spherical coordinates.

Cylindrical Coordinates are given by,

$(x, y, z) = (r \cos \theta, r \sin \theta, z)$

Here,

$r =5.38$

And,

$\theta = 21.8^{\circ}$

By substituting the values, we get,

$(x, y, z) = (20.2, 8.09, 3)$

Spherical Coordinates are given by,

$(x, y, z) = (\rho \sin \phi \cos \theta, \rho \sin \phi \sin \theta, \rho \cos \phi)$

We calculated the values of $r$ and $\theta$ above and now we calculate $\rho$ and $\phi$ for spherical coordinates.

$\rho = r^2 + z^2$

$\rho = 6.16$

We know that $\phi$ is the angle between $\rho$ and $z-axis$, and by using geometry we know that $\phi$ is also the angle between $\rho$ and the vertical side of the right-angled triangle.

$\phi = 90^{\circ} – \theta$

$\phi = 68.2^{\circ}$

By substituting the values and implying, we get:

$(x, y, z) = (5.31, 2.12, 2.28)$

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