# 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`

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

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

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### Mass Density

- $\rho$

Used to denote the **(mass) density** of a given body:

- $\rho = \dfrac m V$

where:

### Area Mass Density

- $\rho_A$

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

- $\rho_A = \dfrac m A$

where:

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

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### Electric Charge Density

- $\map \rho {\mathbf r}$

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 \(\map \rho {\mathbf r}\) is `\map \rho {\mathbf r}`

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### 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`

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### 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.

### Radius Vector

- $\rho$

Used to denote the length of the **radius vector** of a given point in spherical coordinates.