apparent contradiction to Poincaré-Hopf

Gary
GaryVasco

Posts: 3,352

apparent contradiction to Poincaré-Hopf Jan 10, 2018 5:48:52 GMT

Quote

Post by Gary on Jan 10, 2018 5:48:52 GMT

Vasco,

The attachment poses a problem arising from an apparent contradiction of the Poincaré-Hopf Theorem. Since I have no illusions of upsetting centuries of established math, I am wondering what is wrong with my view of the problem.

sphere_saddle.pdf (593.85 KB) (edited following Vasco's response below)

Gary

Last Edit: Jan 10, 2018 16:46:26 GMT by Gary

Admin Administrator Posts: 3,655	apparent contradiction to Poincaré-Hopf Jan 10, 2018 8:38:16 GMT Quote Select Post Deselect Post Link to Post Member Give Gift Back to Top Post by Admin on Jan 10, 2018 8:38:16 GMT Gary Your sphere has 4 vortices and 2 simple crosspoints and so the sum of all the indices on the sphere is $4\times 1+2\times(-1)=4-2=2$ which agrees with the theorem. Vasco
	Last Edit: Jan 10, 2018 9:18:32 GMT by Admin

Gary
GaryVasco

Posts: 3,352

apparent contradiction to Poincaré-Hopf Jan 10, 2018 15:46:51 GMT

Quote

Post by Gary on Jan 10, 2018 15:46:51 GMT

Jan 10, 2018 8:38:16 GMT Admin said:

Gary

Your sphere has 4 vortices and 2 simple crosspoints and so the sum of all the indices on the sphere is

$4\times 1+2\times(-1)=4-2=2$

which agrees with the theorem.

Vasco

Vasco,

Thank you. That dispels my confusion. Ah, I see on p. 463, line 4 that Needham wrote "sum the indices of the singular points." I think I misdirected my attention to the vortices.

Gary

Last Edit: Jan 10, 2018 16:00:46 GMT by Gary

Visual Complex Analysis

apparent contradiction to Poincaré-Hopf

Post by Gary on Jan 10, 2018 5:48:52 GMT

Post by Admin on Jan 10, 2018 8:38:16 GMT

Post by Gary on Jan 10, 2018 15:46:51 GMT

Quick Reply