# Symbols:Greek/Delta

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## Delta

The $4$th letter of the Greek alphabet.

Minuscule: $\delta$
Majuscule: $\Delta$

The $\LaTeX$ code for $\delta$ is \delta .

The $\LaTeX$ code for $\Delta$ is \Delta .

### Diagonal Relation

$\Delta_S$

Let $S$ be a set.

The diagonal relation on $S$ is a relation $\Delta_S$ on $S$ such that:

$\Delta_S = \set {\tuple {x, x}: x \in S} \subseteq S \times S$

Alternatively:

$\Delta_S = \set {\tuple {x, y}: x, y \in S: x = y}$

The $\LaTeX$ code for $\Delta_S$ is \Delta_S .

### Product of Differences

$\Delta_n \left({x_1, x_2, \ldots, x_n}\right)$

Let $n \in \Z_{> 0}$ be a strictly positive integer.

Let $\tuple {x_1, x_2, \ldots, x_n}$ be an ordered $n$-tuple of real numbers.

The product of differences of $\tuple {x_1, x_2, \ldots, x_n}$ is defined and denoted as:

$\map {\Delta_n} {x_1, x_2, \ldots, x_n} = \displaystyle \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}$

When the underlying ordered $n$-tuple is understood, the notation is often abbreviated to $\Delta_n$.

Thus $\Delta_n$ is the product of the difference of all ordered pairs of $\tuple {x_1, x_2, \ldots, x_n}$ where the index of the first is less than the index of the second.

The $\LaTeX$ code for $\Delta_n \left({x_1, x_2, \ldots, x_n}\right)$ is \Delta_n \left({x_1, x_2, \ldots, x_n}\right) .

### Kronecker Delta

$\delta_{x y}$

Let $\Gamma$ be a set.

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Then $\delta_{\alpha \beta}: \Gamma \times \Gamma \to R$ is the mapping on the cartesian square of $\Gamma$ defined as:

$\forall \left({\alpha, \beta}\right) \in \Gamma \times \Gamma: \delta_{\alpha \beta} := \begin{cases} 1_R & :\alpha = \beta \\ 0_R & :\alpha \ne \beta \end{cases}$

This use of $\delta$ is known as the Kronecker delta notation or Kronecker delta convention.

It can be expressed in Iverson bracket notation as:

$\delta_{\alpha \beta} := \left[{\alpha = \beta}\right]$

The $\LaTeX$ code for $\delta_{x y}$ is \delta_{x y} .

### Change

$\Delta x_n$

$\Delta$ is often used to mean change or difference.

For example, for the definition of slope:

$\dfrac {\Delta y} {\Delta x} = \dfrac {y_2 - y_1} {x_2 - x_1} = \dfrac {\text{change in } y} {\text{change in } x}$

The $\LaTeX$ code for $\Delta x_n$ is \Delta x_n .