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- When n = 3, the level set is called a level surface. As the graph of a function f (x, y, z) of three variables is a set (called hypersurface) in R 4 — hence, their graphs cannot be represented— the level surfaces are the only way to graphically represent a function of three variables.
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Example 1: The graph of $z=f(x,\,y)$ as a surface in $3$-space can be regarded as the level surface $w = 0$ of the function $w(x,\,y,\,z) = z - f(x,\, y)$. Example 2: Spheres $x^2+y^2+z^2 = r^2$ can be interpreted as level surfaces $w = r^2$ of the function $w = x^2+y^2+z^2$.
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A level set of a function of two variables f(x, y) f (x, y) is a curve in the two-dimensional xy x y -plane, called a level curve. A level set of a function of three variables f(x, y, z) f (x, y, z) is a surface in three-dimensional space, called a level surface.
When n = 3, a level set is called a level surface (or isosurface); so a level surface is the set of all real-valued roots of an equation in three variables x1, x2 and x3. For higher values of n, the level set is a level hypersurface, the set of all real-valued roots of an equation in n > 3 variables.
Level surface are basically the same as level curves in principle, except that the domain of f(x, y, z) f (x, y, z) is in 3D-space. Therefore, the set f(x, y, z) = k f (x, y, z) = k describes a surface in 3D-space rather than a curve in 2D-space. The following diagram shows the level surfaces.
When $n=3$, the level set is called a level surface. As the graph of a function $f(x,y,z)$ of three variables is a set (called hypersurface) in $\mathbb{R}^4$— hence, their graphs cannot be represented— the level surfaces are the only way to graphically represent a function of three variables.
4.1.2 Sketch a graph of a function of two variables. 4.1.3 Sketch several traces or level curves of a function of two variables. 4.1.4 Recognize a function of three or more variables and identify its level surfaces.
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A quadric surface is a level surface of a second degree polynomial Q ( x, y) . Indeed, the sphere of radius R centered at the origin is a level surface of level k = R2 of the second degree polynomial. Moreover, a sphere is a special type of ellipsoid, which is a surface of the form.
