# Level quantity

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

the *level of* the function for the level or level .

If there is an order relation (with inverse relation ), we can define the following terms.

The *sub-level* amount is the amount

referred to in the case is .

The amount becomes the *super level* amount

referred to in the case is .

## 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.

## 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

and the super level amount too

- .

## Individual evidence

- ^ 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).