Level set

In mathematics, a level set of a real-valued function f of n variables is a set of the form

{ (x₁,...,x_n) | f(x₁,...,x_n) = c }

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), if it is three this is a level surface, and for higher values of n the level set is a level hypersurface.

More specifically, a level curve is the set of all real-valued roots of an equation in two variables x₁ and x₂. A level surface is the set of all real-valued roots of an equation in three variables x₁, x₂ and x₃. A level hypersurface is the set of all real-valued roots of an equation in n (n > 3) variables.

A set of the form

{ (x₁,...,x_n) | f(x₁,...,x_n) ≤ c }

is called a sublevel set of f.

Alternative names

Level curves of a trefoil knot. Red curves are closest to the viewer, while yellow curves are farthest.

A rendered trefoil knot in a different orientation.

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, isobars, isotherms, isogons and isochrones are isocontours.

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

Lastly, a general level set is also called a fiber.

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

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.

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. Let x₀ be the point of interest. The level set going through x₀ is {x | f(x) = f(x₀)}. Consider a curve γ(t) in the level set going through x₀, so we will assume that γ(0) = x₀. We have

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

Equivalently, the Jacobian of f at x₀ is the gradient at x₀

Thus, the gradient of f at x₀ is perpendicular to the tangent γ′(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 of f.

See also

• Isosurface
• Contour line
• Level set method
• Level set (data structures)
• Gradient descent
• Constraint (mathematics)
• Implicit function
• Metaballs
• HyperFun

References

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