Reproduced from William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Problems (Hoboken: John Wiley, 2005), 35.
Gradient vector field of along with its contour map, courtesy of a beautiful hack by Thomas Sefzick.
At increasing levels of detail, the surface approaches its tangent plane around P.
Tangent at a point of plane-ellipsoid intersection:
Family of curves
Curve with contour
Context from #math on freenode:
00:57 <klutometis> I've come up with a new nomination: the "erotic paraboloid" 00:57 <klutometis> http://wikisophia.org/wiki/User:Danenberg#Erotic_paraboloid 00:58 <thermoplyae> you appear to have a lot of free time 00:59 <klutometis> thermoplyae: me? it corresponds roughly with calculus; so no expenditure of free time 00:59 <TRWBW> klutometis: hehe, n1, something just like it at http://www.youtube.com/watch?v=eBGIQ7ZuuiU 01:00 <thermoplyae> you say that, but all the busy people i know don't draw graphs of "co-sinusoidal demi-toroid"s 01:00 <klutometis> damn it, TRWBW; I can't believe I got rick rolled 01:00 <thermoplyae> just sayin' 01:00 <thermoplyae> haha 01:01 <klutometis> thanks, boys; good night