Two is easy, four is elegant

Sometimes the more sophisticated stuff is easier.

Each section of the school year has its challenges for our tutor.  Autumn sees a new crop of students battling with trigonometry, some of which requires insight, some just memorization.  Summer is normally given over to preparations for the standardized tests.  As midwinter fades into the rear-view mirror, there appear the most advanced topics in the most advanced classes; and often our tutor has forgotten how to do them.

Part of the trouble is that he has few students working at this level, and some years none at all; so he sees the topics only briefly and at long intervals.  And part of the trouble is that, while the more elementary topics are now routine, and he has worked (and published) using higher-level techniques than ever appear in tutoring sessions, most of these springtime topics lie in his dusty undergraduate notes from ancient days.

Let’s be more specific.  He has lots of students dealing with the calculus of a curve on a two-dimensional plane.  There are plenty of devious questions to ask in this subject, but it’s a familiar enough part of the world that it normally presents no surprises.  At the other end of his experience lie the four-dimensional vectors of Special Relativity and differential forms, plus the single theorem that unites Stokes and Gauss in any number of dimensions.  In between is the space of three dimensions.

In three dimensions, you have to draw your figures carefully and watch how your coordinates run.  In two, it’s not hard; in four, you use tensors.  The equation of a line in two dimensions is a Middle School skill; in four, the geodesic equation is employed.  The equations of lines and planes in 3D are easier than the latter, but our tutor only uses them for a week every year or two.  In two dimensions, one rarely changes coordinate systems; in four (or more) it’s routine, and again one trots out the tensors.  In 3D the explicit transformation from one to another is tedious and a source of errors.

It’s also something of a source of frustration that, while Calculus in three dimensions is introduced, there isn’t time to do very much with it.  The equations of fluid dynamics and of electromagnetic fields are just about within the grasp of our tutor’s most advanced students.  But they never actually work with them.  Not until college, anyway.

To be fair, the students are very patient, and allow our tutor extra time to look things up and work out the details.  And the details normally come back quickly.  Maybe his irritation is mostly due to the reminder of how much he once knew, that is now mostly forgotten.