Wronskian

From Academic Kids

In mathematics, the Wronskian is a function named after Polish mathematician Josef Hoene-Wronski, especially important in the study of differential equations.

Given a set of n functions f1, ..., fn, the Wronskian W(f1, ..., fn) is given by:

<math> W(f_1, \ldots, f_n) = \begin{vmatrix} f_1 & f_2 & \cdots & f_n \\ f_1' & f_2' & \cdots & f_n' \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)} & f_2^{(n-1)} & \cdots & f_n^{(n-1)} \end{vmatrix} <math>

That is, it is the determinant of the matrix constructed by placing the functions in the first row, the first derivative of each function in the second row, and so on through the n-1 derivative, thus forming a square matrix.

The Wronskian can be used to determine whether the set of functions is linearly independent. If the Wronskian is non-zero at some point in a region, then the functions are linearly independent in that region. Note that if the Wronskian is zero at every point in a region, the functions may either be linearly independent or linearly dependent.

Abstract definition

There is a sense in which the Wronskian of a n-th order linear differential equation is its n-th exterior power. Of course for that idea to be implemented one must be working with some formulation in which differential equations are sufficiently like vector spaces: for example in the language of vector bundles carrying a connection.pl:Wrońskian

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