Function Conjunction → The Anatomy of a Physical Expression

Factors serve as The Anatomy of a Physical Expression. They come in several types as listed below, each characterized as having a distinct role in defining a property of a physical system. The following list items are partially underlined to make memorization easy:

Constants
Coefficients
Quantities
Proximities
Dislocations
Directions

Definition

A Physical Expression
Constant (or 1)	$\times$
Coefficient (or 1)	$\times$
Quantity (or 1)	$\times$
Proximity (or 1)	$\times$
Dislocation (or 1)	$\times$
Direction (or 1)	$=$

Constants

$c$ = Speed of Light
$G$ = Gravitational constant
$k_B$ = Boltzmann's constant
$\alpha$ = Fine Structure constant
$\mu_0$ = Magnetic Permeability of Free Space
$\epsilon_0$ = Electric Permittivity of Free Space

Coefficients

$\mu_r$ = Relative Magnetic Permeability
$\epsilon_r$ = Relative Electric Permittivity

Quantities

$q$ = point charge
$\lambda_q$ = linear charge density (for continuous charge)
$\sigma_q$ = surface charge density (for continuous charge)
$\rho_q$ = volume charge density (for continuous charge)
$m$ = mass
$\rho$ = volume mass density

Proximities

$\frac{1}{|\mathbf{r}-\mathbf{r'}|}$ = inverse of the magnitude of the separation between positions $\mathbf{r}$ and $\mathbf{r'}$
$\frac{1}{|\mathbf{r}-\mathbf{r'}|^2}$ = inverse square of the magnitude of the separation between positions $\mathbf{r}$ and $\mathbf{r'}$

Dislocations

$\mathbf{x}$ = position
$\mathbf{v}$ = velocity
$\mathbf{a}$ = acceleration

Dislocations according to an inertial observer at time $t$

$\mathbf{r}$ = position of a charge $q$ at time $t$ , when it receives a light signal from $q'$ that was emitted earlier at the retarded time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$
$\frac{∂\mathbf{r}}{∂t}$ = $\mathbf{\dot{r}}$ = velocity of a charge $q$ at time $t$ , when it receives a light signal from $q'$ that was emitted earlier at the retarded time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$
$\frac{∂^2\mathbf{r}}{∂t^2}$ = $\mathbf{\ddot{r}}$ = acceleration of a charge $q$ at time $t$ , when it receives a light signal from $q'$ that was emitted earlier at the retarded time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$
$\mathbf{r'}$ = position a charge $q'$ had at the retarded time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$ , when it emitted a light signal which has now reached $q$ at position $\mathbf{r}$ and time $t$
$\frac{∂\mathbf{r'}}{∂t}$ = $\mathbf{\dot{r}'}$ = velocity a charge $q'$ had at the retarded time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$ , when it emitted a light signal which has now reached $q$ at position $\mathbf{r}$ and time $t$
$\frac{∂^2\mathbf{r'}}{∂t^2}$ = $\mathbf{\ddot{r}'}$ = acceleration a charge $q'$ had at the retarded time $t' = t - |\mathbf{r}-\mathbf{r'}|/c$ , when it emitted a light signal which has now reached $q$ at position $\mathbf{r}$ and time $t$

Directions

$\mathbf{\hat{x}}$ = position unit vector
$\mathbf{\hat{v}}$ = velocity unit vector
$\mathbf{\hat{a}}$ = acceleration unit vector

Directions according to an inertial observer at time $t$

$\mathbf{\hat{r}}$ = position unit vector of $q$ at time $t$
$\mathbf{\hat{\dot{r}}}$ = velocity unit vector of $q$ at time $t$
$\mathbf{\hat{\ddot{r}}}$ = acceleration unit vector of $q$ at time $t$
$\mathbf{\hat{r'}}$ = position unit vector of $q'$ at retarded time $t'$
$\mathbf{\hat{\dot{r'}}}$ = velocity unit vector of $q'$ at retarded time $t'$
$\mathbf{\hat{\ddot{r'}}}$ = acceleration unit vector of $q'$ at retarded time $t'$