The Wronskian is a useful and standard object for testing the linear dependence of a set of differentiable functions ${y_i (x)}$. Are there similar useful ways for testing the linear dependence of a set of differentiable functions with two variables ${y_i (x1, x2)}$ (for instance, i= 1,2,3)? I searched on Google Scholar, but it is unclear what the most common or simplest tool is.
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1$\begingroup$ In principle, a generalization of the Wronskian can be defined. You consider all partial derivatives up to order $n$, let us say there are $d(n)$ of them, and you compute those for $d(n)$ functions. Some properties of the usual Wronskian probably carry over. For example, if the $d(n)$ functions are linearly dependent, then the Wronskian vanishes identically. This is the easy direction. In order to obtain results in the other direction, one needs to impose conditions (I haven't thought long about it). $\endgroup$Malkoun– Malkoun2025-02-10 04:33:14 +00:00Commented Feb 10 at 4:33
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2$\begingroup$ Generalized Wronskians have been studied, and the W=0 => lin. dep. implication established in some special cases, see doi.org/10.1016%2F0024-3795%2889%2990548-X. By the way, the link to this paper can be found on the Wikipedia page about the Wronskian. $\endgroup$Andrei Smolensky– Andrei Smolensky2025-02-10 12:41:03 +00:00Commented Feb 10 at 12:41
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