Level quantity

Sets of levels (black lines) around a saddle point of a function of two variables

In mathematics , a designated level set or Level amount of the set of all points of the definition range of a function , which is assigned to a same function value. Closely related terms for functions with values ​​in an ordered set are those of the sub-level set , which contains all points whose function values ​​do not exceed a specified value, and the super-level set , which contains all points whose function values ​​do not fall below a specified value.

definition

Let there be sets , a function and a value from the target set , then is called ${\displaystyle U,V}$ ${\displaystyle f\colon U\to V}$ ${\displaystyle c\in V}$

${\displaystyle {\mathcal {N}}_{f}(c):=f^{-1}(c)=\{x\in U\mid f(x)=c\}\subseteq U}$

the level of the function for the level or level . ${\displaystyle f}$ ${\displaystyle c}$

If there is an order relation (with inverse relation ), we can define the following terms. ${\displaystyle V}$ ${\displaystyle \leq }$ ${\displaystyle \geq }$

The sub-level amount is the amount

${\displaystyle {\mathcal {L}}_{f}^{\leq }(c):=\{x\in U\mid f(x)\leq c\}}$

referred to in the case is . ${\displaystyle V=\mathbb {R} }$ ${\displaystyle {\mathcal {L}}_{f}^{\leq }(c)=f^{-1}(\left(-\infty ,c\right])}$

The amount becomes the super level amount

${\displaystyle {\mathcal {L}}_{f}^{\geq }(c):=\{x\in U\mid f(x)\geq c\}}$

referred to in the case is . ${\displaystyle V=\mathbb {R} }$ ${\displaystyle {\mathcal {L}}_{f}^{\geq }(c)=f^{-1}(\left[c,\infty \right))}$

Applications

physics

For two-dimensional scalar fields , a level set is usually a line and one speaks of an isoline or level line. For three-dimensional scalar fields (for example for scalar potential fields ) this set is mostly a curved surface and it is called isosurface or level surface (e.g. contour lines ).
The term level surface is also used for force fields such as the electric field or magnetic fields .

Economics

For a production function and a production level is the amount of all bundles of production factors with which the amount can be generated. The amount is referred to as the isoquant to the production level. ${\displaystyle f\colon (0,\infty )^{n}\to (0,\infty )}$ ${\displaystyle c\in (0,\infty )}$ ${\displaystyle {\mathcal {N}}_{f}(c)=f^{-1}(c)}$ ${\displaystyle c}$ ${\displaystyle {\mathcal {N}}_{f}(c)}$ ${\displaystyle c}$

generalization

If the function is real-vector-valued , i.e. if it has the image space and if this is provided with a generalized inequality , the sub-level set can be generalized ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \preccurlyeq _{K}}$

${\displaystyle {\mathcal {L}}_{f}^{\leq }(c):=\{x\in U\mid f(x)\preccurlyeq _{K}c\}}$

and the super level amount too

${\displaystyle {\mathcal {L}}_{f}^{\geq }(c):=\{x\in U\mid f(x)\succcurlyeq _{K}c\}}$.

Individual evidence

1. ^ Klaus D. Schmidt: Mathematics . Basics for economists. 2nd Edition. Springer-Verlag, Berlin 2000, ISBN 978-3-540-66521-2 , p. 369 ( limited preview in Google Book search).