How to define a level set in higher dimension As the topic, how to define level set?? I know it is about level curve, what about when it comes to n dimension
 A: Given a differentiable function $(x,y)\mapsto f(x,y)$ and and a value $c\in{\mathbb R}$ one can define the level set
$$\gamma:=f^{-1}(c)=\bigl\{(x,y)\in{\rm dom}(f)\ \bigm|\ f(x,y)=c\bigr\}\ .$$
This is the set of all points $(x,y)\in{\rm dom}(f)$ where $f(x,y)=c$. 
"Generically" the set $\gamma$ is a curve in the $(x,y)$-plane. To be exact: If $(x_0,y_0)\in\gamma\ $ and $\nabla f(x_0,y_0)\ne(0,0)$, say $f_y(x_0,y_0)\ne0$, then in the neighborhood of $(x_0,y_0)$ the set $\gamma$ is the graph of a local function $y=\phi(x)$.
Now in $n$ dimensions we are given a differentiable function $${\bf x}=(x_1,x_2,\ldots, x_n)\mapsto f(x_1,x_2,\ldots, x_n)\ .$$ Again, for any given value $c\in{\mathbb R}$ we can define the level set
$$L_c:=f^{-1}(c)=\bigl\{{\bf x}\in{\rm dom}(f)\ \bigm|\ f({\bf x})=c\bigr\}$$
consisting of all points in ${\rm dom}(f)$ where $f$ assumes the value $c$.
"Generically" the set $L_c$ is a manifold of dimension $n-1$ (also called a hypersurface) in ${\mathbb R}^n$. 
To be exact: If ${\bf x}_0$ is a point of $L_c\ $ and if $\nabla f({\bf x}_0)\ne{\bf 0}$, say ${\partial f\over\partial x_n}({\bf x}_0)\ne0\ $, then in the neighborhood of ${\bf x}_0$ the set $L_c$ is the graph of a local function  $x_n=\phi(x_1,x_2,\ldots, x_{n-1})$ of $n-1$ variables. Intuitively speaking: Near ${\bf x}_0$ the level set $L_c$ can be viewed as a landscape over the $(x_1,\ldots, x_{n-1})$-plane.
When the "technical" condition $\nabla f({\bf x}_0)\ne{\bf 0}$ is not fulfilled at the point ${\bf x}_0\in L_c$ then one has to expect that the manifold character of $L_c$  is defect at ${\bf x}_0$. E.g., $L_c$ might consist only of the isolated point ${\bf x}_0$, as is the case for the function $f({\bf x}):=\|x\|^2$ at ${\bf x}_0:={\bf 0}$.
A: A level set by definition is the inverse image of a point in the range. For example, if  $f(x,y) = x^2 + y^2$, then the level set corresponding to $1$ is the set $\{(x,y): f(x,y) = 1\}$ which is just the collection of pairs for which $x^2 + y^2 = 1$, i.e. the unit circle.
