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Level curve

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Level curve

For the computational technique, see Level set method.
For level surfaces of force fields, see equipotential surface.
Points at constant slices of x2 = f(x1).
Lines at constant slices of x3 = f(x1, x2).
Planes at constant slices of x4 = f(x1, x2, x3).
(n − 1)-dimensional level sets for functions of the form f(x1, x2, ..., xn) = a1x1 + a2x2 + ... + anxn where a1, a2, ..., an are constants, in (n + 1)-dimensional Euclidean space, for n = 1, 2, 3.
Points at constant slices of x2 = f(x1).
Contour curves at constant slices of x3 = f(x1, x2).
Curved surfaces at constant slices of x4 = f(x1, x2, x3).
(n − 1)-dimensional level sets of non-linear functions f(x1, x2, ..., xn) in (n + 1)-dimensional Euclidean space, for n = 1, 2, 3.

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

L_c(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) = c \right\}~,

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

When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline. When n = 3, a level set is called a level surface (see also isosurface), and for higher values of n the level set is a level hypersurface.

A set of the form

L_c^-(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) \leq c \right\}

is called a sublevel set of f (or, alternatively, a lower level set or trench of f).

L_c^+(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) \geq c \right\}

is called a superlevel set of f.[1][2]

A level set is a special case of a fiber.


Properties

  • The gradient of f at a point is perpendicular to the level set of f at that point.
  • The sublevel sets of a convex function are convex (the converse is however not generally true).

See also

References

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