# Symbols:Greek/Kappa

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## Kappa

The $10$th letter of the Greek alphabet.

Minuscule: $\kappa$
Majuscule: $\Kappa$

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

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

### Kernel

Used in abstract algebra and other related fields to denote the general kernel of a homomorphism.

## Definition

Let $C$ be a curve defined by a real function which is twice differentiable.

The curvature of a $C$ is the reciprocal of the radius of the osculating circle to $C$

### Whewell Form

The curvature $k$ of $C$ at a point $P$ can be expressed in the form of a Whewell equation as:

$k = \dfrac {\mathrm d \psi} {\mathrm d s}$

where:

$\psi$ is the turning angle of $C$
$s$ is the arc length of $C$.

### Cartesian Form

Let $C$ be embedded in a cartesian plane.

The curvature $k$ of $C$ at a point $P = \left({x, y}\right)$ is given by:

$k = \dfrac {y''} {\left({1 + y'^2}\right)^{3/2} }$

where:

$y' = \dfrac {\mathrm d y} {\mathrm d x}$ is the derivative of $y$ with respect to $x$ at $P$
$y'' = \dfrac {\mathrm d^2 y} {\mathrm d x^2}$ is the second derivative of $y$ with respect to $x$ at $P$.

### Parametric Form

Let $C$ be embedded in a cartesian plane and defined by the parametric equations:

$\begin{cases} x = x \left({t}\right) \\ y = y \left({t}\right) \end{cases}$

The curvature $k$ of $C$ at a point $P = \left({x, y}\right)$ is given by:

$k = \dfrac {x' y'' - y' x''} {\left({x'^2 + y'^2}\right)^{3/2} }$

where:

$x' = \dfrac {\mathrm d x} {\mathrm d t}$ is the derivative of $x$ with respect to $t$ at $P$
$y' = \dfrac {\mathrm d y} {\mathrm d t}$ is the derivative of $y$ with respect to $t$ at $P$
$x''$ and $y''$ are the second derivatives of $x$ and $y$ with respect to $t$ at $P$.