Level set

	Email		Print
	Bookmark		Link

Level set

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a level set of a real

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
-valued function

Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
f of n variables is a set of the form

where c is a constant. That is, it is the set where the function takes on a given constant value.

When the number of variables is two, this is a level curve (contour line

Contour line

A contour line of a Function of two variables is a curve along which the function has a constant value. In cartography, a contour line joins points of equal elevation above a given level, such as mean sea level....
), if it is three this is a level surface, and for higher values of n the level set is a level hypersurface
Hypersurface

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface....
.

More specifically, a level curve is the set of all real-valued roots of an equation in two variables x₁ and x₂.

Discussion

Ask a question about 'Level set'

Start a new discussion about 'Level set'

Answer questions from other users

Full Discussion Forum

Encyclopedia

In mathematics

Mathematics

Real number

Function (mathematics)

where c is a constant. That is, it is the set where the function takes on a given constant value.

When the number of variables is two, this is a level curve (contour line

Contour line

is called a sublevel set of f.

Alternative names

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

Implicit function

In mathematics, an implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable....
. The name isocontour is also used, which means a contour of equal height. In various applications, isobar

Isobar

Isobar may refer to:* a contour line of equal or constant pressure in meteorology* two nuclides with the same mass number in nuclear physics* a heat pipe...
s, isotherm

Isotherm

An isotherm may refer to:*A type of contour line or surface connecting points of equal temperature*An isothermal process in a thermodynamic cycle....
s, isogons and isochrones are isocontours.

Analogously, a level surface is sometimes called an implicit surface or an isosurface
Isosurface

An isosurface is a dimension analog of an isocontour. It is a surface that represents points of a constant value within a volume of space; in other words, it is a level set of a continuous function whose domain is 3D-space....
.

Lastly, a general level set is also called a fiber
Fiber (mathematics)

In mathematics, the fiber of a point y under a function f : X ? Y is the inverse relation of under f, that is, ...
.

Example

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

r²=x² + y²

If we choose r=5 then our isovalue is c=5²=25.

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

Theorem
Theorem

In mathematics, a theorem is a statement Mathematical proof on the basis of previously accepted or established statements such as axioms.In formal mathematical logic, the concept of a theorem may be taken to mean a formula that can be formal proof according to the deductive system of a fixed formal system....
. The gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of f at a point is perpendicular to the level set of f at that point.

This theorem is quite remarkable. To understand what it 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...

Proof
Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive reasoning or empirical arguments....
. Let x₀ be the point of interest. The level set going through x₀ is . Consider a curve ?(t) in the level set going through x₀, so we will assume that ?(0) = x₀. We have

Now let us differentiate at t = 0 by using the chain rule

Chain rule

In calculus, the chain rule is a formula for the derivative of the functional composition of two function .In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of Mathematics#Change of y with respect to x can be computation as the rate of chan...
. We find

Equivalently, the Jacobian

Jacobian

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.In algebraic geometry the Jacobian of a algebraic curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded....
of f at x₀ is the gradient at x₀

Thus, the gradient of f at x₀ is perpendicular to the tangent ?′(0) to the curve (and to the level set) at that point. Since the curve ?(t) is arbitrary, it follows that the gradient is perpendicular to the level set. Q.E.D.

Q.E.D.

Q.E.D. is an abbreviation of the List of Latin phrases , which literally means "which was to be demonstrated". The phrase is written in its abbreviated form at the end of a mathematical proof or Philosophy Logical argument, to signify that the last statement deduced was the one to be demonstrated, so the proof is complete....

A consequence of this theorem is that if a level set crosses itself (more precisely, fails to be a smooth submanifold

Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties....
or hypersurface

Hypersurface

In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface....
) then the gradient vector must be zero at all points of crossing. Then, every point in the crossing will be a critical point

Critical point

Critical point may refer to:*Critical point *Critical point *Critical point See also*Brillouin zone*Percolation thresholds...
of f.