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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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For a function of three variables, a level set is a surface in three-dimensional space that we will call a level surface. For a constant value c c in the range of f(x, y, z) f (x, y, z), the level surface of f f is the implicit surface given by the graph of c = f(x, y, z) c = f (x, y, z).
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.
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.
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.
Level surfaces are represented mathematically as the set of points satisfying the equation $f(x, y, z) = c$, where $c$ is a constant. Each level surface corresponds to a specific value of the function and can reveal insights about the function's behavior and its critical points.
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Given a function of 3 variables U ( x, y, z) , we define the level surface of U ( x, y, z) of level k to be the set of all points in R3 which are solutions to. U ( x, y, z) = k. Indeed, many of the most familiar surfaces are level surfaces of functions of 3 variables.
