# 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

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Did you try Wikipedia? – lhf Apr 9 '12 at 10:57
Question is vague? A $c$-level set of a function $f:\mathbb{R}^n \to \mathbb{R}$ is the set $\{ p \in \mathbb{R}^n : f(p) = c \}.$ Example: In 3D, the surface of the unit sphere is the $1$-level set of the function $f : \mathbb{R}^3 \to \mathbb{R},$ defined as $f : (x,y,z) \mapsto x^2 + y^2 + z^2.$ – user2468 Apr 9 '12 at 18:11

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 ${\bf x}_0:={\bf 0}$.

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nice answer sir (+1) – funktor Apr 9 '12 at 13:13

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.

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