Notation

General

$f(\mathbf{x};\mathbf{p})$ is a quantity $f$ with variables $\mathbf{x}$ and (optionally) parameterized by parameters $p$

Quantity	Notation	Subscript Left	Subscript Right	Superscript	Example
Scalar	lowercase	n/a	n/a	n/a	$a=4$
Vector	lowercase, bold	frame	point (from) → point (to)	n/a	${}_{\mathcal{A}}{\mathbf{r}}_{AB}$
Unit axis vector	lowercase, bold, non-italic	frame	axis	frame of axis	${}_{\mathcal{A}}{\mathbf{e}}_x^{\mathcal{B}}$
Homogeneous vector	lowercase, bold, tilde	frame	point (from) → point (to)	n/a	${}_{\mathcal{A}}{\tilde{\mathbf{r}}}_{AB}$
Matrix	uppercase, bold	n/a	n/a	n/a	$\mathbf{A}$
Transformation matrix	uppercase, bold, non-italic	n/a	frame (to) ← frame (from)	n/a	${\mathbf{R}}_{\mathcal{B}\mathcal{A}}$

Kinematics

Quantity	Notation	Subscript
Absolute position	${\mathbf{r}}_P:={}_{\mathcal{I}}{\mathbf{r}}_{IP}$	object (point)
Absolute velocity	${\mathbf{v}}_P:={}_{\mathcal{I}}{\dot{\mathbf{r}}}_{IP}$	object (point)
Absolute acceleration	${\mathbf{a}}_P:={\dot{\mathbf{v}}}_P={}_{\mathcal{I}}{\ddot{\mathbf{r}}}_{IP}$	object (point)
Note: absolute means the quantity is expressed wrt a fixed (inertial) reference frame $\mathcal{I}$ with origin $I$

Probability

Property	Notation
Random variable (RV), state	$X$, $x$
Probability	$P(X=x)=:P(X)$
Conditional probability	$P(X=x|Y=y)=:P(X|Y)$
Expectation of a continuous RV	${\mathbb{E}}_{x\sim f(x)}[X]={\int }_{-\infty }^{\infty }xf(x)dx=:\mathbb{E}[X]=:\mu$
Expectation of a discrete RV	${\mathbb{E}}_{x\sim p(x)}[X]={\sum }_i{x}_{i}p(x_i)=:\mathbb{E}[X]=:\mu$
Expectation (continuous RV) of a function $g$	${\mathbb{E}}_{x\sim f(x)}[g(x)]={\int }_{-\infty }^{\infty }g(x)p(x)dx$
Expectation (discrete RV) of a function $g$	${\mathbb{E}}_{x\sim p(x)}[g(x)]={\sum }_{i}g(x_i)p(x_i)$
Variance	$\text{Var}[X]=:{\sigma }^2$
Standard deviation	$\text{SD}[X]=:\sigma$
Probability mass function (for discrete RVs)	$p_X(x)=:p(x)$
Probability density function (for continuous RVs)	$f_X(x)=:f(x)$
Cumulative distribution function	$F_X(x)=:F(x)$