3-D Coordinate Systems

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Cartesian Coordinates

Here is a figure showing the definitions of the three Cartesian coordinates $(x,y,z)$

$cart1.svg$

and here are three figures showing a surface of constant $x\text{,}$ a surface of constant $x\text{,}$ and a surface of constant $z\text{.}$

$cart3.svg$ $cart4.svg$ $cart2.svg$

Finally here is a figure showing the volume element $\mathrm{d}V$ in cartesian coordinates.

$cart5.svg$

Cylindrical Coordinates

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

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

$cyl1 (1).svg$

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 $z\text{.}$

$cyl3.svg$ $cyl4.svg$ $cyl2.svg$

Finally here is a figure showing the volume element $\mathrm{d}V$ in cylindrical coordinates.

$cyl5.svg$

Spherical Coordinates

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

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

$spherical.svg$

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

$sphericalSide.svg$ $sphericalTop.svg$

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 $\varphi\text{.}$

$spher2.svg$ $spher3.svg$ $spher4.svg$

Finally, here is a figure showing the volume element $\mathrm{d}V$ in spherical coordinates

$spher5.svg$

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

$spher6.svg$ $spher7.svg$

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