# Symbols:Greek/Rho

## 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:

### Area Density

- $\rho_A$

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

- $\rho_A = \dfrac m A$

where:

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

.

### Volume Charge Density

- $\rho$

Used to denote volume charge density:

- $\rho = \dfrac {\d Q} {\d V}$

where:

- $Q$ is the electric charge
- $V$ is the volume

### 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.