A.6: 3-D Coordinate Systems

Last updated
Save as PDF

Page ID: 92254

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

A.6.1 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.

$cart5.svg$

A.6.2 Cylindrical Coordinates

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

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

$cyl1 (1).svg$

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.

$cyl5.svg$

A.6.3 Spherical Coordinates

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

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

$spherical.svg$

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

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

Here are three figures showing a surface of constant $\rho\text{,}$ a surface of constant $\theta\text{,}$ and a surface of constant $\vec{a}rphi\text{.}$

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

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 $\rho\ \mathrm{d}\vec{arphi}$ and $\rho\sin\vec{a}rphi\ \mathrm{d}{\theta} $ arise.

$spher6.svg$ $spher7.svg$

Search

Text Color

Text Size

Margin Size

Font Type