User dettonville - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T16:28:52Z http://mathoverflow.net/feeds/user/20528 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85490/implicit-function-theorem-at-a-singular-point Implicit function theorem at a singular point? dettonville 2012-01-12T14:11:56Z 2012-01-13T13:14:36Z <p>Let $F:\mathbb{R}^2 \rightarrow \mathbb{R}$ be three times continuously differentiable in some open neighborhood $\mathcal{U}$ of $(0,0)$. Suppose that $F(0,0) = F_x(0,0) = F_y(0,0) = F_{xy}(0,0) = 0$ and that $F_{yy} \not = 0$ and $F_{xx}(0,0)/F_{yy}(0,0) &lt; 0$.</p> <p>Obviously I cannot apply the implicit function theorem. Under which circumstances can I still express y as a function of x locally aoround (0,0)? It seems as though it should be $y'(0) = \pm \sqrt{-\frac{F_{xx}(0,0)}{F_{yy}(0,0)}}$...</p> <p>EDIT: Fixed typo.</p> http://mathoverflow.net/questions/85490/implicit-function-theorem-at-a-singular-point/85529#85529 Comment by dettonville dettonville 2012-01-13T08:34:08Z 2012-01-13T08:34:08Z Wow, thanks so much. I can't believe you even did all the epsilons and deltas! Working through it now, looks like I'll learn Morse Theory some other time. http://mathoverflow.net/questions/85490/implicit-function-theorem-at-a-singular-point Comment by dettonville dettonville 2012-01-12T15:35:02Z 2012-01-12T15:35:02Z Yes, that was a typo. Thanks, I'm just now reading up on morse theory. http://mathoverflow.net/questions/85490/implicit-function-theorem-at-a-singular-point/85498#85498 Comment by dettonville dettonville 2012-01-12T15:23:41Z 2012-01-12T15:23:41Z Thanks so much, that's exactly what I was looking for! And your're right of course, it should be $F_{xx}/F_{yy}&lt;0$.