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About thirty years ago, in a practice GRE (Graduate Record Exam) math test in the US, a question assumed the student knew the definition of the Wronskian. I had never heard of this determinant before.

Q. Is this still assumed to be part of what a student heading toward a graduate degree in mathematics knows?

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    $\begingroup$ Well, it is very relevant in the basic not-purely-algorithmic theory of second-order ordinary differential equations... So, if someone didn't have any recognition of that name/label, I'd think they'd not studied differential equations beyond the very-most-basic. $\endgroup$
    – paul garrett
    May 19, 2022 at 23:49
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    $\begingroup$ In Europe it would be universally expected that a student beginning graduate studies in mathematics knew what the Wronskian was. This would normally be taught in the first or second year of undergraduate studies. In the US standards vary considerably and it is possible to graduate with a degree in mathematics without studying ODEs - but such a student is not prepared for graduate study. Such a student may survive anyway, but that is a different matter. $\endgroup$
    – Dan Fox
    May 20, 2022 at 7:09
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    $\begingroup$ I don't know if it is still true. But thirty years ago, the final five questions of the GRE Math 2 exam were deliberately on advanced and obscure topics that was not assumed to be in the domain of every test taker. The example question of this sort that sticks with me was the formal definition of continuity in an arbitrary topological space. I suppose it was their stance that not everyone who took the test was entitled to an 800, and I think there is an argument to be made for that. Anyways, I got an 800 on the test in 1989 and the definition of Wronskian was not part of my prep at all. $\endgroup$
    – Matthew Daly
    May 20, 2022 at 21:42
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    $\begingroup$ "but such a student is not prepared for graduate study." - yeah, strongly disagree here, though I see where you are coming from. Of the triumvirate of Algebra, Analysis, and Topology/Geometry that dominates quals (or whatever the equivalent is at various institutions, we had coursework instead), ODE doesn't directly show up in any of those (though of course it is related to all three in various ways). (Whether a graduating physics major should leave without seeing a decent ODE class is a separate matter.) $\endgroup$
    – kcrisman
    May 23, 2022 at 20:58
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    $\begingroup$ We'll have to agree to disagree on this. Certainly all your examples are well chosen, as is the historical basis. But what actual analysts, topologists, geometers ... do on a daily basis may or may not have any direct connection to it. (Pun not intended, but enjoyed.) A student who knows no algebra is, indeed, badly prepared for graduate study in (pure) math. $\endgroup$
    – kcrisman
    May 24, 2022 at 12:21

2 Answers 2

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I would say the assumption is that people heading to mathematics graduate school know about the Wronskian, but this assumption isn't universally true.

Certainly, anyone who has studied a semester of differential equations (and is heading to graduate school) should know it.

However, there is a substantial minority of people interested in pure mathematics of a generally algebraic bent who never study differential equations. (I was one of these people until I was assigned to teach it!)

There really isn't anything that's universally expected (rather than merely generally assumed) of people heading to mathematics graduate school other than an ability to read and write proofs, and some folks doing mathematical modelling might even disagree with that.

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    $\begingroup$ Agreed with your assessment for algebraically minded students. I once taught such a student DEqns in about 4 days. 4 painful, mind-bending days. But, it's really just algebra for the most part. $\endgroup$
    – James S. Cook
    May 20, 2022 at 1:29
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    $\begingroup$ Agreed. My advisor didn't allow me to take an ODE course, as there was nothing in it that you couldn't learn from speed-reading any intro textbook over the course of a weekend. $\endgroup$
    – Matthew Daly
    May 20, 2022 at 21:51
  • $\begingroup$ This is perhaps true in the US (although that is debatable) but it is certainly false in most of Europe. In most of Europe curricula in undergraduate math degrees are well-defined nationally and all of them include ordinary differential equations. It is difficult to teach even an introductory nonrigorous ODE course for engineering students without mentioning the Wronskian (if one does not teach the basics of solving homogeneous linear ODE's, what does one teach in such a course?) $\endgroup$
    – Dan Fox
    Jun 23, 2022 at 9:44
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It is discussed in all the introductory DEqns texts of which I've used. It's needed to complete the discussion of linear independence of solution sets. Together with Abel's formula it provides some rather general theorems for linear n-th order ODEs. The Wronskian lies at the heart of variation of parameters which is the general method to solve nonhomogeneous n-th order ODEs.

It's also a bit tricky since the theorems which hold for solution sets do not hold for arbitrary sets of functions. In particular, the Wronksian can vanish identically for linearly independent functions. But, that can never happen for a solution set thanks to Abel's formula.

Incidentally, all these things generalize to calculus over a finite dimensional commutative algebra.

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