How to check whether a subset of R^2 is a smooth curve

Question

Is there any useful lemma for me know whether a subset of R^2 is a smooth curve. Is $S=\{(x,y)\in \mathbb{R}^2:|x|^8+|y|^8=1\}$ a smooth curve? What about $T=\{(x,y)\in \mathbb{R}^2:|x|^{\frac{1}{2}}+|y|^{\frac{1}{2}}=1\}$?Is there a general method to check?

Answer 1

You are considering curves $f(x,y)=C$. By the implicit function theorem, you can find a smooth patch of the curve around any point where the gradient of the equation exists and is different from the zero vector.

For $f(x,y)=x^8+y^8=1$ the gradient is $(8x^7,8y^7)$ which is zero in both components only at the origin, which is not on the curve.

For $f(x,y)=\sqrt{|x|}+\sqrt{|y|}=1$ the gradient is not defined at points $(x,y)=(\pm1,0)$ and $(0,\pm1)$. Inspection shows that you get cusps at these points, which are not parts of a smooth curve.