Definition:Wronskian/General Definition

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 $f_1$.