Symbols:Greek/Delta
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 arbitrarily small change or difference in the value of the (real) variable $x$.
For example, for the definition of derivative:
- $\ds \dfrac {\d y} {\d x} = \lim_{\delta x \mathop \to 0} \dfrac {\delta y} {\delta x} = \lim_{x_2 - x_1 \mathop \to 0} \dfrac {y_2 - y_1} {x_2 - x_1} = \lim_{\text{change in } x \mathop \to 0} \dfrac {\text{change in } y} {\text{change in } x}$
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:
$\quad \forall \tuple {\alpha, \beta} \in \Gamma \times \Gamma: \delta_{\alpha \beta} := \begin{cases} 1_R & : \alpha = \beta \\ 0_R & : \alpha \ne \beta \end{cases}$
This use of $\delta$ is known as the Kronecker delta notation or Kronecker delta convention.
The $\LaTeX$ code for \(\delta_{x y}\) is \delta_{x y} .
Kronecker Delta (Tensor Form)
- ${\delta^i}_j$
When used in the context of tensors, the notation for the Kronecker delta can often be seen as ${\delta^i}_j$.
The $\LaTeX$ code for \({\delta^i}_j\) is {\delta^i}_j .
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:
- $\map \delta x := \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$
The $\LaTeX$ code for \(\map \delta x\) is \map \delta x .
Declination
- $\delta$
The symbol used to denote declination is $\delta$.
The $\LaTeX$ code for \(\delta\) is \delta .
Diagonal Relation
- $\Delta_S$
Let $S$ be a set.
The diagonal relation on $S$ is the relation $\Delta_S$ on $S$ defined as:
- $\Delta_S = \set {\tuple {x, x}: x \in S} \subseteq S \times S$
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:
- $\map {\Delta_n} {x_1, x_2, \ldots, x_n} = \ds \prod_{1 \mathop \le i \mathop < j \mathop \le n} \paren {x_i - x_j}$
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 .
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Appendix $1$: Symbols and Conventions: Greek Alphabet