We had this definition for a regular surface given:
We say that a set $M=\left\{x\in \mathbb{R}^n\::\:\Psi ^{\nu }\left(x\right)=0,\:\nu =1,\:...,\:n\right\}$ where $\Psi \:^{\nu \:}:\:\mathbb{R}^n\:\rightarrow \:\mathbb{R}$ are smooth functions defines a regular surface if $rank\frac{\partial \left(\Psi ^1\left(x\right),..,\Psi \:^s\left(x\right)\right)\:}{\partial \left(x^1,..,x^s\right)\:}=s$ (for $x\in M$)
Could someone maybe give me an example to understand this definition intuitively?