Level set
In mathematics, a level set of a function is a set of the form
- { (x0,...,xn) | f(x0,...,xn) = c }
where c is constant. That is, it is the set where a function of several variables 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 a level hypersurface.
The gradient vector at a given point is perpendicular to the level set at that point, {x | f(x) = f(x0)}. If we have some smooth parametrization of the set x(t), with x(0)=x0 then
- f(x(0)) = f(x0) = k.
Now differentiating at t=0, using the chain rule
- 0 = (Jx0 f)(x ′(0))
Equivalently, the Jacobian of f at x0 is the gradient at x0
- 0 = (∇f)(x0) · x ′(0)
So x ′ (0) is perpendicular to (∇f)(x0), and similarly to the level set of x0.
A consequence 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 a point of crossing. This then will be a critical point of f.
See also contour line.
