Editing Level set (section)

{{Short description|Subset of a function's domain on which its value is equal}}
{{for|the computational technique|Level-set method}}
{{redirect|Level surface|the application to force fields|Equipotential surface|water level surfaces|Equigeopotential}}
{{Use American English|date = April 2019}}

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

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

In [[mathematics]], a '''level set''' of a [[real-valued function]] {{mvar|f}}  of {{mvar|n}} [[Function of several real variables|real variables]] is a [[set (mathematics)|set]] where the function takes on a given [[constant (mathematics)|constant]] value {{mvar|c}}, that is:

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

When the number of independent variables is two, a level set is called a '''level curve''', also known as ''[[contour line]]'' or ''isoline''; so a level [[curve]] is the set of all real-valued solutions of an equation in two variables {{math|''x''{{sub|1}}}} and {{math|''x''{{sub|2}}}}.  When {{math|1=''n'' = 3}}, a level set is called a '''level surface''' (or ''[[isosurface]]''); so a level [[Surface (mathematics)|surface]] is the set of all real-valued roots of an equation in three variables {{math|''x''{{sub|1}}}}, {{math|''x''{{sub|2}}}} and {{math|''x''{{sub|3}}}}. For higher values of {{mvar|n}}, the level set is a '''level hypersurface''', the set of all real-valued roots of an equation in {{math|''n'' > 3}} variables (a [[higher-dimensional]] [[hypersurface]]).

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