Level set

A level set of a function is the set of inputs at which the function equals a specified value. Its shape and dimension depend on the function and the value chosen.

Function: A rule that assigns each input in a domain exactly one output. A level set collects inputs by their shared output under a function.
Regular level set theorem: A theorem stating that the preimage of a regular value is an embedded submanifold. It gives precise conditions for a level set to be a smooth manifold.
Implicit surface: A surface defined as the solution set of an equation involving its coordinates. A three-dimensional level set commonly forms an implicitly defined surface.
Fiber (mathematics): The preimage of a point under a map between mathematical spaces. A level set is a fiber when the function's output is fixed to one value.
Inverse image: The set of inputs mapped by a function into a specified subset of its codomain. A level set is the inverse image of a single output value.
Tangent space: A vector space of directions tangent to a manifold at one of its points. At a regular point, its vectors lie in the kernel of the function's derivative.
Level-set method: A numerical method that tracks moving interfaces as level sets of a higher-dimensional function. It uses a function's level set to represent and evolve a boundary.
Zero set: The set of points where a function equals zero. It is the special level set selected by the value zero.
Contour line: A curve joining points on a plane where a scalar field has one constant value. For a function of two variables, a level set is often a contour line.
Coarea formula: An integration formula relating integrals over a domain to integrals over level sets. It turns calculations across a region into integrals over its level sets.

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