Let $f\in C^{2}(\mathbb{R}^2,\mathbb{R})$ such that $\nabla f(x)\neq 0$ for all $x$. Define a plane foliation $\{ \sum_{\bar{x}}^{f}\; ; x\in\mathbb{R}^2 \}$, where $\sum_{\bar{x}}^{f}=\{ x\in\mathbb{R}^2:f(x)=f(\bar{x}) \}$ are the level sets of $f$. Show that:
- the connected components of $\sum_{\bar{x}}^{f}$ are smooth curves;
- such components are parameterized by $\gamma(t)=(x_1(t),x_2(t))$ with $\gamma(0)=\bar{x}$ and $$ x_1'(t)=(\partial_{x_2}f)(x_1(t),x_2(t)), $$ $$ x_2'(t)=-(\partial_{x_1}f)(x_1(t),x_2(t)). $$
- there are smooth one-dimensional foliations of $\mathbb{R}^{2}\backslash\{(0,0)\}$ that are not given by foliations $\sum_{\bar{x}}^{f}$ to some smooth $f$?
I wasn't able to develop this exercise much, so any help would be good. Thanks in advance! :D