I have difficulty understanding the following Theorem(adsbygoogle = window.adsbygoogle || []).push({});

If U is open in [itex]ℝ^2[/itex], [itex]F: U \rightarrow ℝ[/itex] is a differentiable function with Lipschitz derivative, and [itex]X_c=\{x\in U|F(x)=c\}[/itex], then [itex]X_c[/itex] is a smooth curve if [itex][\operatorname{D}F(\textbf{a})][/itex] is onto for [itex]\textbf{a}\in X_c[/itex]; i.e., if [tex]\big[ \operatorname{D}F\bigl( \begin{smallmatrix}a \\ b\end{smallmatrix}\bigr)\big]≠0 \mbox{ for all } \textbf{a}=\bigl( \begin{smallmatrix}a \\ b \end{smallmatrix}\bigr)\in X_c [/tex]

I don't understand why the differential of F at a being onto is equivalent to saying the differential is not zero. Can someone explain? Thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Level sets as smooth curves

**Physics Forums | Science Articles, Homework Help, Discussion**