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CLP-4 Vector Calculus

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 \(\varphi\text{.}\)
Here is a figure showing the surface element \(\dee{S}\) 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.
Finally, here is a figure showing the volume element \(\dee{V}\) in spherical coordinates