3d Coordinate Systems

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Section A.6 3d Coordinate Systems

Subsection A.6.1 Cartesian Coordinates

Here is a figure showing the definitions of the three Cartesian coordinates

(x, y, z)

and here are three figures showing a surface of constant

x,

a surface of constant

x,

and a surface of constant

z .

Finally here is a figure showing the volume element

d V

in cartesian coordinates.

Subsection A.6.2 Cylindrical Coordinates

Here is a figure showing the definitions of the three cylindrical coordinates

\begin{aligned} r & = distance from (0, 0, 0) to (x, y, 0) \\ θ & = angle between the x axis and the line joining (x, y, 0) to (0, 0, 0) \\ z & = signed distance from (x, y, z) to the x y -plane \end{aligned}

The cartesian and cylindrical coordinates are related by

\begin{aligned} x & = r \cos θ & y & = r \sin θ & z & = z \\ r & = \sqrt{x^{2} + y^{2}} & θ & = \arctan \frac{y}{x} & z & = z \end{aligned}

Here are three figures showing a surface of constant

r,

a surface of constant

θ,

and a surface of constant

z .

Finally here is a figure showing the volume element

d V

in cylindrical coordinates.

Subsection A.6.3 Spherical Coordinates

Here is a figure showing the definitions of the three spherical coordinates

\begin{aligned} ρ & = distance from (0, 0, 0) to (x, y, z) \\ φ & = angle between the z axis and the line joining (x, y, z) to (0, 0, 0) \\ θ & = angle between the x axis and the line joining (x, y, 0) to (0, 0, 0) \end{aligned}

and here are two more figures giving the side and top views of the previous figure.

The cartesian and spherical coordinates are related by

\begin{aligned} x & = ρ \sin φ \cos θ & y & = ρ \sin φ \sin θ & z & = ρ \cos φ \\ ρ & = \sqrt{x^{2} + y^{2} + z^{2}} & θ & = \arctan \frac{y}{x} & φ & = \arctan \frac{\sqrt{x^{2} + y^{2}}}{z} \end{aligned}

Here are three figures showing a surface of constant

ρ,

a surface of constant

θ,

and a surface of constant

φ .

Finally, here is a figure showing the volume element

d V

in spherical coordinates

and two extracts of the above figure to make it easier to see how the factors

ρ d φ

and

ρ \sin φ d θ

arise.