Symbols:Greek/Rho

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Rho

The $17$th letter of the Greek alphabet.

Minuscules: $\rho$ and $\varrho$

Majuscule: $\Rho$

The $\LaTeX$ code for \(\rho\) is \rho .
The $\LaTeX$ code for \(\varrho\) is \varrho .

The $\LaTeX$ code for \(\Rho\) is \Rho .

Density

$\rho$

Used to denote the density of a given body:

$\rho = \dfrac m V$

where:

$m$ is the body's mass

$V$ is the body's volume

Area Density

$\rho_A$

Used to denote the area density of a given two-dimensional body:

$\rho_A = \dfrac m A$

where:

$m$ is the body's mass

$A$ is the body's area.

The $\LaTeX$ code for \(\rho_A\) is \rho_A .

Electric Charge Density

$\rho$

Let $A$ be a point in space in which an electric field acts.

Let $\delta V$ be a volume element containing $A$.

The (electric) charge density $\map \rho {\mathbf r}$ at $A$ is defined as:

\(\ds \map \rho {\mathbf r}\)

\(=\)

\(\ds \lim_{\delta V \mathop \to 0} \dfrac Q {\delta V}\)

\(\ds \)

\(=\)

\(\ds \dfrac {\d Q} {\d V}\)

where:

$Q$ denotes the electric charge within $\delta V$

$\mathbf r$ denotes the position vector of $A$.

Thus the electric charge density is the quantity of electric charge per unit volume, at any given point in that volume:

The $\LaTeX$ code for \(\rho\) is \rho .

Right Regular Representation

$\rho_a$

Let $\struct {S, \circ}$ be an algebraic structure.

The mapping $\rho_a: S \to S$ is defined as:

$\forall x \in S: \map {\rho_a} x = x \circ a$

This is known as the right regular representation of $\struct {S, \circ}$ with respect to $a$.

The $\LaTeX$ code for \(\map {\rho_a} x\) is \map {\rho_a} x .

Radius of Curvature

$\rho$

The radius of curvature of a curve $C$ at a point $P$ is defined as the reciprocal of the absolute value of its curvature:

$\rho = \dfrac 1 {\size k}$

Autocorrelation

$\rho_k$

Let $S$ be a stochastic process giving rise to a time series $T$.

The autocorrelation of $S$ at lag $k$ is defined as:

$\rho_k := \dfrac {\expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} } } {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }$

where:

$z_t$ is the observation at time $t$

$\mu$ is the mean of $S$

$\expect \cdot$ is the expectation.

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