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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 \(x\text{,}\) a surface of constant \(x\text{,}\) and a surface of constant \(z\text{.}\)
Finally here is a figure showing the volume element \(\dee{V}\) in cartesian coordinates.
Subsection 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 $x$ 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 \(\dee{V}\) in cylindrical coordinates.
Subsection 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 \(\varphi\text{.}\)
Finally, here is a figure showing the volume element \(\dee{V}\) in spherical coordinates
and two extracts of the above figure to make it easier to see how the factors \(\rho\ \dee{\varphi}\) and \(\rho\sin\varphi\ \dee{\theta}\) arise.
