EDB — 1GF

Exercises

1. [1GF] Consider the following \(C^∞\) function of 2 variables

   \[ {f}(x,y)= x^ 3+y^ 4-1 \quad . \]

   Check that \(\{ f=0\} =\{ (x,y)∈ℝ^ 2: f(x,y)=0\} \) is not empty; then, for each point of the plane where \(f\) vanishes, discuss whether the implicit function theorem can be applied, and therefore if the set \(\{ f=0\} \) is locally graph of a \(C^∞\) function. Also study the set \(\{ f=0\} \): is it compact? How many connected components are there?

   (Please note what is shown in [1H3]↺↻).

   Solution 1

   [1GG]↺↻