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.
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.15$: The General Solution of the Homogeneous Equation: Theorem $\text{A}$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Wronskian
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Wronskian
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Wronskian