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{{for|the computational technique|Level set method}}
{{for|'''level surfaces''' of [[Force field (physics)|force fields]]|equipotential surface}}

{{multiple image
|width=140
|image1=Level sets linear function 2d.svg
|caption1=Points at constant slices of {{nowrap|''x''<sub>2</sub> {{=}} ''f''(''x''<sub>1</sub>)}}.
|image2=Level sets linear function 3d.svg
|caption2=Lines at constant slices of {{nowrap|''x''<sub>3</sub> {{=}} ''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>)}}.
|image3=Level sets linear function 4d.svg
|caption3=Planes at constant slices of {{nowrap|''x''<sub>4</sub> {{=}} ''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}}.
|footer={{nowrap|(''n'' − 1)}}-dimensional level sets for functions of the form {{nowrap|''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) {{=}} ''a''<sub>1</sub>''x''<sub>1</sub> + ''a''<sub>2</sub>''x''<sub>2</sub> + ... + ''a''<sub>''n''</sub>''x''<sub>''n''</sub>}} where ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> are constants, in {{nowrap|(''n'' + 1)}}-dimensional Euclidean space, for ''n'' = 1, 2, 3.}}

{{multiple image
|width=140
|image1=Level sets non-linear function 2d.svg
|caption1=Points at constant slices of {{nowrap|''x''<sub>2</sub> {{=}} ''f''(''x''<sub>1</sub>)}}.
|image2=Level sets non-linear function 3d.svg
|caption2=Contour curves at constant slices of {{nowrap|''x''<sub>3</sub> {{=}} ''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>)}}.
|image3=Level sets non-linear function 4d.svg
|caption3=Curved surfaces at constant slices of {{nowrap|''x''<sub>4</sub> {{=}} ''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}}.
|footer={{nowrap|(''n'' − 1)}}-dimensional level sets of non-linear functions ''f''(''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) in {{nowrap|(''n'' + 1)}}-dimensional Euclidean space, for ''n'' = 1, 2, 3.}}

In [[mathematics]], a '''level set''' of a [[real number|real]]-valued [[function of several real variables|function of ''n'' real variables]] ''f'' is a set of the form

: <math> L_c(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) = c \right\}~, </math>

that is, a set where the function takes on a given constant value ''c''.

When the number of variables is two, a level set is generically a curve, called  a level curve, [[contour line]], or isoline.  So a level curve is the set of all real-valued solutions of an equation in two variables ''x''<sub>1</sub> and ''x''<sub>2</sub>.   When ''n''&nbsp;=&nbsp;3, a level set is called a level surface (see also [[isosurface]]), and for  higher values of ''n'' the level set is a level hypersurface.  So a ''level surface'' is the set of all real-valued roots of an equation in three variables ''x''<sub>1</sub>, ''x''<sub>2</sub> and ''x''<sub>3</sub>, and a level hypersurface is the set of all real-valued roots of an equation in ''n'' (''n'' > 3) variables.

A level set is a special case of a [[fiber (mathematics)|fiber]].

{{-}}

==Alternative names==

[[Image:trefoil knot level curves.png|thumb|Intersections of a [[co-ordinate]] function's level surfaces with a [[trefoil knot]].  Red curves are closest to the viewer, while yellow curves are farthest.]]

Level sets show up in great many applications, often under different names. 

For example, a level curve is also called an implicit curve, emphasizing that such a curve is defined by an [[implicit function]]. The name isocontour is also used, which means a contour of equal height. In various applications, [[isobar (meteorology)|isobar]]s, [[isotherm (contour line)|isotherm]], [[isogon]]s and [[isochrone]]s are isocontours.

Analogously, a level surface is sometimes called an mplicit surface or an [[isosurface]].

== Example ==

For example, given a specific radius ''r'', the equation of a circle defines an isocontour.

<math>r^2=x^2 + y^2</math>

If we choose <math>r=5</math> then our isovalue is <math>c=5^2=25</math>.

All points (x,y) that evaluate to 25 constitute the isocontour. This means that they are a member of the isocontour's level set. If a point evaluates to less than 25 the point is on the inside of the isocontour. If the result is greater than 25, it is on the outside.

==Level sets versus the gradient==

[[Image:level grad.svg|right|thumb|300px|Consider a function ''f'' whose graph looks like a hill. The blue curves are then the level sets. The red curves follow the direction of the gradient. In other words, the cautious hiker follows the blue paths, while the bold one the red paths.]]

'''[[Theorem]].''' The [[gradient]] of ''f'' at a point is perpendicular to the level set of ''f'' at that point. 

To understand what this means, imagine that two hikers are at the same location on a mountain. One of them is bold, and decides to go in the direction where the slope is steepest. The other one is more cautious; he does not want to either climb or descend, choosing a path which will keep him at the same height. In our analogy, the above theorem says that the two hikers will depart in directions perpendicular to one another.

'''[[Mathematical proof|Proof]].''' Let <math>\mathbf x_0</math> be the point of interest and <math>f(x_0)=c</math>. The level set going through <math>\mathbf x_0</math> is <math>\Phi_c = \{ \mathbf x  : f(\mathbf x) =c \}</math>. Consider for some <math>\delta>0</math> a curve  <math>\mathbf \gamma(t):(-\delta,\delta)\to \Phi_c</math> such that <math>\mathbf \gamma(0)= \mathbf x_0 </math>.
We have

: <math>\forall t\in (-\delta,\delta): f(\mathbf \gamma(t)) = c .</math>

Now let us differentiate at ''t'' = 0 by using the [[chain rule]]. We find

:<math>J_f({\mathbf x_0}) {\mathbf \gamma}'(0)=0.</math>

Equivalently, the [[Jacobian]] of ''f'' at '''x'''<sub>0</sub> is the gradient at '''x'''<sub>0</sub> 

:<math>\nabla f({\mathbf x}_0) \cdot {\mathbf \gamma}'(0)=0.</math>

Thus, the gradient of ''f'' at '''x'''<sub>0</sub> is perpendicular to the tangent  '''γ'''&prime;(0) to the curve (and to the level set) at that point. Since the curve  '''γ'''(''t'') is arbitrary, it follows that the gradient is perpendicular to the level set. [[Q.E.D.]]

A consequence of this theorem is that if a level set crosses itself (more precisely, fails to be a smooth [[submanifold]] or [[hypersurface]]) then the gradient vector must be zero at all points of crossing. Then, every point in the crossing  will be a [[critical point (mathematics)|critical point]] of ''f''.

==Sublevel and superlevel sets==
A set of the form

: <math> L_c^-(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) \leq c \right\} </math>

is called a sublevel set of ''f'' (or, alternatively, a lower level set or trench of ''f''). 
: <math> L_c^+(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) \geq c \right\} </math>

is called a superlevel set of ''f''.<ref>{{eom|first=M.I.|last=Voitsekhovskii|id=L/l058220}}</ref><ref>{{MathWorld|title=Level Set|urlname=LevelSet}}</ref> Sublevel sets are important in [[mathematical optimization|minimization theory]]. The [[totally bounded set|boundness]] of some [[empty set|non-empty]] sublevel set and the lower-semicontinuity of the function implies that a function attains its minimum, by [[Extreme_value_theorem#Extension_to_semi-continuous_functions|Weierstrass's theorem]]. The [[convex set|convexity]] of all the sublevel sets characterizes [[quasiconvex function]]s.<ref>{{cite article|last=Kiwiel|first=Krzysztof C.|title=Convergence and efficiency of subgradient methods for quasiconvex minimization|journal=Mathematical Programming  (Series A)|publisher=Springer|location=Berlin, Heidelberg|issn=0025-5610|pages=1-25|volume=90|issue=1|doi=10.1007/PL00011414|doi=10.1007/PL00011414|year=2001|mr=1819784}}</ref>


== See also ==
* [[Epigraph (mathematics)|Epigraph]]
* [[Level set method]]
* [[Level set (data structures)]]

==References==
{{Reflist}}

[[Category:Multivariable calculus]]