# Definition:Wronskian

## Definition

Let $\map f x$ and $\map g x$ be real functions defined on a closed interval $\closedint a b$.

Let $f$ and $g$ be differentiable on $\closedint a b$.

The Wronskian of $f$ and $g$ is defined as:

$\map W {f, g} = \begin {vmatrix} \map f x & \map g x \\ \map {f'} x & \map {g'} x \\ \end {vmatrix} = \map f x \, \map {g'} x - \map g x \, \map {f'} x$

### General Definition

Let $\map {f_1} x, \map {f_2} x, \dotsc, \map {f_n} x$ be real functions defined on a closed interval $\closedint a b$.

Let $f_1, f_2, \ldots, f_n$ be $n - 1$ times differentiable on $\closedint a b$.

The Wronskian of $f_1, f_2, \ldots, f_n$ on $\closedint a b$ is defined as:

$\map W {f_1, f_2, \dotsc, f_n} = \begin {vmatrix} \map {f_1} x & \map {f_2} x & \cdots & \map {f_n} x \\ \map { {f_1}'} x & \map { {f_2}'} x & \cdots & \map { {f_n}'} x \\ \vdots & \vdots & \ddots & \vdots \\ \map { {f_1}^{\paren {n - 1} } } x & \map { {f_2}^{\paren {n - 1} } } x & \cdots & \map { {f_n}^{\paren {n - 1} } } x \\ \end{vmatrix}$

where:

$\begin{vmatrix} \cdots \end{vmatrix}$ denotes the determinant
$\map { {f_1}^{\paren {n - 1} } } x$ denotes the $n - 1$th derivative of $f_1$.

## Also known as

Some sources preserve the diacritic on the n, that is: Wrońskian, but many consider such refinements to be visual clutter and prefer to discard them.

## Source of Name

This entry was named for Józef Maria Hoene-Wroński.