Wronskian Determinant

Definition

Wronskian of 2 functions $W(y_1,y_2)=\left|\begin{array}{cc}{y}_1& y_2\\ y_1^{\prime }& y_2^{\prime }\end{array}\right|=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }$

Let $y_1,\dots ,y_n$ be (n-1)-times differentiable function on $I$
The Wronskian of $y_1,\dots ,y_n$ is defined by
$W(y_1,\dots ,y_n)=\left|\begin{array}{cccc}{y}_1& y_2& \dots & y_n\\ y_1^{\prime }& y_2^{\prime }& \dots & y_n^{\prime }\\ ⋮& ⋮& \ddots & ⋮\\ y_1^{(n-1)}& y_2^{(n-1)}& \dots & y_n^{(n-1)}\end{array}\right|$: function on $I$

Wronskian of column vectors ${\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}$ $W({\mathbf{x}}^{(1)},\dots ,{\mathbf{x}}^{(n)}):=\left|\begin{array}{ccc}{\mathbf{x}}^{(1)}& \dots & {\mathbf{x}}^{(n)}\end{array}\right|=|\mathbf{X}(t)|$