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.

Expert Answer:

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.

Figure-1 : Cartesian Coordinate System

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

1) Cylindrical Coordinates are given by:

$(x, y, z) = (r \cos θ, r \sin θ, z) 0 \leq θ \leq 2 π$

Where,

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

Figure-2 : Cylindrical Coordinate System

Given,

$z = x$

So the equation becomes,

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

2) Spherical Coordinates are given by:

$(x, y, z) = (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ) ρ \geq 0, 0 \leq θ \leq 2 π, 0 \leq ϕ \leq π$

Figure-3 : Spherical Coordinate System

Given,

$z = x$

$ρ \cos ϕ = ρ \sin ϕ \cos θ$

$\frac{\cos ϕ}{\sin ϕ} = \cos θ$

$\cot ϕ = \cos θ$

$θ = \arccos (\cot ϕ)$

By substituting the values we get,

$(x, y, z) = (ρ \sin ϕ \cos (\arccos (\cot ϕ)), ρ \sin ϕ \sin (\arccos (\cot ϕ)), ρ \cos ϕ)$

Simplifying by using trigonometric identities, we get:

$(x, y, z) = (ρ \cos ϕ, ρ \sin ϕ \sqrt{1 - \cot^{2} ϕ}, ρ \cos ϕ)$

Numerical Results:

Cylindrical Coordinates,

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

Spherical Coordinates,

$(x, y, z) = (ρ \cos ϕ, ρ \sin ϕ \sqrt{1 - \cot^{2} ϕ}, ρ \cos ϕ)$

Example:

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

Cylindrical Coordinates are given by,

$(x, y, z) = (r \cos θ, r \sin θ, z)$

Here,

$r = 5.38$

And,

$θ = {21.8}^{\circ}$

By substituting the values, we get,

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

Spherical Coordinates are given by,

$(x, y, z) = (ρ \sin ϕ \cos θ, ρ \sin ϕ \sin θ, ρ \cos ϕ)$

We calculated the values of $r$ and $θ$ above and now we calculate $ρ$ and $ϕ$ for spherical coordinates.

$ρ = r^{2} + z^{2}$

$ρ = 6.16$

We know that $ϕ$ is the angle between $ρ$ and $z - a x i s$ , and by using geometry we know that $ϕ$ is also the angle between $ρ$ and the vertical side of the right-angled triangle.

$ϕ = 90^{\circ} - θ$

$ϕ = {68.2}^{\circ}$

By substituting the values and implying, we get:

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

Images/Mathematical drawings are created with Geogebra.

Expert Answer:

Numerical Results:

Example:

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