Thursday, February 28, 2013

Index Theorem the shape of the figure, namely whether it has three or four vertices.

Index Theorem

When you face a difficult problem try to solve another more simple and related. So instead of

finding the solutions of the differential equations we must seek for how many there are. But an

unexpected result is the format of the answer: we get not a number but a solution expressed in

terms of a geometrical shape. That is quite unusual for our students.

A simple analogue can be looked at triangles and quadrangles in the plane. It can be

complicated to find the angles in the corners of some of these figures, but sometimes, before

Euclid, someone realised that the sum of the angles in all the corners is always 180 degrees for

a triangle, and 360 degrees for a quadrangle. The answer to this question is thus easily given,

and depends only on the shape of the figure, namely whether it has three or four vertices.

No comments:

Post a Comment