Mathematical objects

Example Meaning LaTeX
$P$ Points and positions are denoted by capital italic letters. \$P\$
$(4, 5, -2)$ Coordinates of a position are given as a tuple, so $P$ is at $(4, 5, -2)$ is the same as saying that $P$ has coordinates $x = 4$, $y = 5$, $z = -2$. Note the distinction from vector components with square brackets. \$(4, 5, -2)\$
$\boldsymbol{v}$ Vectors in typeset material are in bold font. \$\boldsymbol{v}\$
$\vec{v}$ Vectors in handwriting use an over-arrow. \$\vec{v}\$
$\|\boldsymbol{v}\|$, $v$ Magnitude uses double-bars or a plain letter, so $v = \|\boldsymbol{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}$. \$\|\boldsymbol{v}\|\$, \$v\$
$\hat{\boldsymbol{v}}$ Unit vectors use over-hat, so $\hat{\boldsymbol{v}} = \frac{\boldsymbol{v}}{\|\boldsymbol{v}\|}$. \$\hat{\boldsymbol{v}}\$
$\hat{\boldsymbol{\imath}}$, $\hat{\boldsymbol{\jmath}}$, $\hat{\boldsymbol{k}}$ Cartesian basis vectors, so we write $\boldsymbol{v} = 3\hat{\boldsymbol{\imath}} + \hat{\boldsymbol{\jmath}} + 7\hat{\boldsymbol{k}}$. \$\hat{\boldsymbol{\imath}}\$, \$\hat{\boldsymbol{\jmath}}\$, \$\hat{\boldsymbol{k}}\$
$[3, 1, 7]$ Vector components use square brackets, so we write $[\boldsymbol{v}]_R = [3, 1, 7] = 3\hat{\boldsymbol{\imath}} + \hat{\boldsymbol{\jmath}} + 7\hat{\boldsymbol{k}}$. If the basis is clear then we will write $\boldsymbol{v} = [3, 1, 7]$. \$[3, 1, 7]\$
$[\boldsymbol{v}]_R$ Vector components in basis $R$. Standard basis names are $R$ for Rectangular (Cartesian), $P$ for polar, $C$ for cylindrical, $S$ for spherical. \$[\boldsymbol{v}]_R\$
$v_x, v_y, v_z$ Vector components are in non-bold with subscripts, so $\boldsymbol{v} = [v_x, v_y, v_z] = v_x\,\hat{\boldsymbol{\imath}} + v_y\,\hat{\boldsymbol{\jmath}} + v_z\,\hat{\boldsymbol{k}}$. \$v_x, v_y, v_z\$
$v$ versus $v_x$ Magnitude (positive) is the plain letter $v$, while signed component is $v_x$. \$v\$ versus \$v_x\$
$\hat{\boldsymbol{e}}_r, \hat{\boldsymbol{e}}_\theta$ Polar basis vectors. Maybe we should change this to $\hat{\boldsymbol{r}}, \hat{\boldsymbol\theta}$? \$\hat{e}_r\$, \$\hat{e}_\theta\$
$\boldsymbol{r}$, $\boldsymbol{r}_P$, $\boldsymbol{r}_{OP}$, $\overrightarrow{OP}$ Position vector of point $P$ from origin $O$. The origin and/or point can be neglected if it is obvious from context. \$\boldsymbol{r}\$, \$\boldsymbol{r}_P\$, \$\boldsymbol{r}_{OP}\$, \$\overrightarrow{OP}\$
$\boldsymbol{\rm A}$ Matrices are in upright (roman) bold. \$\boldsymbol{\rm A}\$
$A_{ij}$ Matrix components are in italic non-bold font. \$A_{ij}\$
$4\rm\ kg/m^2$, $4\rm\ kg\,m^{-2}$ Units are in roman (upright) font, have a space between the number and units, and have a thin-space between unit symbols. \$4\rm\ kg/m^2\$, \$4\rm\ kg\,m^{-2}\$
$x = 4t^2$ To make formulas dimensionally correct we use one of the following forms: (1) “$x = 4t^2$, where $t$ is in seconds and $x$ is in meters”, (2) “$x = a t^2$ where $a = 4\rm\ m/s^2$”, or (3) “$x = 4 (t/{\rm s})^2\rm\ m$” (using quantity calculus). \$x = 4t^2\$

Diagram elements

Element Meaning LaTeX
$\mathcal{B}_1$ Body number 1. Use numbers 1,2,3 for bodies. \$\mathcal{B}_1\$
$m_1, \omega_1, \alpha_1$ Mass, angular velocity, and angular acceleration of body $\mathcal{B}_1$. Use subscript numbers for quantities associated with bodies. \$m_1, \omega_1, \alpha_1\$
$P, Q$ Points $P$ and $Q$. Use italic capital letters for points. \$P, Q\$
$\boldsymbol{r}_P, \boldsymbol{v}_P, \boldsymbol{a}_P$ Position, velocity, and acceleration vectors of point $P$. Use subscript capital italic letters for quantities associated with points. \$\boldsymbol{r}_P, \boldsymbol{v}_P, \boldsymbol{a}_P\$
$I_{1,P,z}$ Moment of inertia of body $\mathcal{B}_1$ about point $P$ around the $z$ axis. Any of the subscripts can be neglected if they are obvious from context, although at least the point should normally be included. \$I_{1,P,z}\$
$\bigoplus$ Center of mass. Sometimes used instead of $C$ if there are too many points on a body or to avoid confusion. \$\bigoplus, \oplus\$

Color scheme for diagrams

When possible, use a pale yellow (or tan) background with a light gray (or light green) square grid, like traditional engineering paper. A blank white background can also be used.

Color Meaning
Black Coordinate axes and objects.
Gray Measurements, angles, other notes.
Blue Position vectors.
Green Velocities and angular velocities.
Cyan Accelerations and angular accelerations.
Red Forces.
Purple Moments.

Example diagram showing colors

Example diagram showing colored elements.