A.6: 3d Coordinate Systems

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 \(x\text{,}\) a surface of constant \(x\text{,}\) and a surface of constant \(z\text{.}\)

                        

Finally here is a figure showing the volume element \(\text{d}V\) in cartesian coordinates.

A.6.2 Cylindrical Coordinates

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

\[\begin{align*} r&=\text{ distance from }(0,0,0)\text{ to }(x,y,0)\\ \theta&=\text{ angle between the the $x$ axis and the line joining $(x,y,0)$ to $(0,0,0)$}\\ z&=\text{ signed distance from }(x,y,z) \text{ to the $xy$-plane} \end{align*}\]

The cartesian and cylindrical coordinates are related by

\[\begin{align*} x&=r\cos\theta & y&=r\sin\theta & z&=z\\ r&=\sqrt{x^2+y^2} & \theta&=\arctan\frac{y}{x} & z&=z \end{align*}\]

Here are three figures showing a surface of constant \(r\text{,}\) a surface of constant \(\theta\text{,}\) and a surface of constant \(z\text{.}\)

                        

Finally here is a figure showing the volume element \(\text{d}V\) in cylindrical coordinates.

A.6.3 Spherical Coordinates

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

\[\begin{align*} \rho&=\text{ distance from }(0,0,0)\text{ to }(x,y,z)\\ \varphi&=\text{ angle between the $z$ axis and the line joining $(x,y,z)$ to $(0,0,0)$}\\ \theta&=\text{ angle between the $x$ axis and the line joining $(x,y,0)$ to $(0,0,0)$} \end{align*}\]

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

                

The cartesian and spherical coordinates are related by

\[\begin{align*} x&=\rho\sin\varphi\cos\theta & y&=\rho\sin\varphi\sin\theta & z&=\rho\cos\varphi\\ \rho&=\sqrt{x^2+y^2+z^2} & \theta&=\arctan\frac{y}{x} & \varphi&=\arctan\frac{\sqrt{x^2+y^2}}{z} \end{align*}\]

Here are three figures showing a surface of constant \(\rho\text{,}\) a surface of constant \(\theta\text{,}\) and a surface of constant \(\varphi\text{.}\)

                        

Here is a figure showing the surface element \(\text{d}S\) in spherical coordinates

and two extracts of the above figure to make it easier to see how the factors \(\rho\ \text{d}\varphi\) and \(\rho\sin\varphi\ \text{d}\theta\) arise.

                        

Finally, here is a figure showing the volume element \(\text{d}V\) in spherical coordinates