Symbols:Delta

Delta

Diagonal Relation

$\Delta_S$

Let $S$ be a set.

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

$\Delta_S = \left\{{\left({x, x}\right): x \in S}\right\} \subseteq S \times S$

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

Product of Differences

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

Let $n \in \Z, n > 0$ be an integer.

Then $\Delta_n \left({x_1, x_2, \ldots, x_n}\right)$ is defined as:

$\displaystyle \Delta_n = \prod_{1 \le i < j \le n} \left({x_i - x_j}\right)$

Thus $\Delta_n$ is the product of the difference of all pairs of $\left\{{x_1, x_2, \ldots, x_n}\right\}$ 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 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.