# 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 .

### Arbitrarily Small Real Number

$\delta$

$\delta$ is often used to mean an arbitrarily small (strictly) positive real number.

For example, the definition of a limit of a real function $f$:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$

is interpreted to mean:

For every arbitrarily small (strictly) positive real number $\epsilon$, there exists an arbitrarily small (strictly) positive real number $\delta$ such that every real number $x \ne c$ in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of $L$.

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

### Arbitrarily Small Change

$\delta x$

$\delta x$ is often used to mean an arbiitrary small change or difference in the value of the (real) variable $x$.

For example, for the definition of derivative:

$\displaystyle \dfrac {\d y} {\d x} = \lim_{\delta x \mathop \to 0} \dfrac {\delta y} {\delta x} = \lim_{x_2 - x_1 \mathop \to 0} \dfrac {y_2 - y_1} {x_2 - x_1} = \lim_{\text{change in } x \mathop \to 0} \dfrac {\text{change in } y} {\text{change in } x}$

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

### 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:

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

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

### Dirac Delta Function

$\map \delta x$

Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

Consider the real function $F_\epsilon: \R \to \R$ defined as:

$\map {F_\epsilon} x := \begin{cases} 0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \\ 0 & : x > \epsilon \end{cases}$

The Dirac delta function is defined as:

$\map \delta x := \displaystyle \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$

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

### 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

$\map {\Delta_n} {x_1, x_2, \ldots, x_n}$

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 $\map {\Delta_n} {x_1, x_2, \ldots, x_n}$ is \map {\Delta_n} {x_1, x_2, \ldots, x_n} .

### Change

$\Delta x$

$\Delta x$ is often used to mean change or difference in the value of the (real) variable $x$.

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 $\Delta x$ is \Delta x .

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