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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 $c$ in the range of $f(x,y,z)$, the level surface of $f$ is the implicit surface given by the graph of $c=f(x,y,z)$.
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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 $w = x^2+y^2+z^2$.
It is difficult to draw many interesting level surfaces by hand, but CalcPlot3D helps us explore them easily. There are actually two ways to enter and graph the level surface equations for a particular function of three variables in CalcPlot3D:
The following diagram shows the level surfaces \[f(x,y,z) = x^2 + y^2 - x^2 = k\] for various \(k\) values. The level surfaces are hyperbolas of one or two sheets, depending on the values of \(k\). Nevertheless, the value of \(f(x,y,z)\) stays the same at each points on a level surface.
Nov 21, 2021 · If you just set the function equal to \(a\), a slider will automatically be created so you can just dynamically observe the level surfaces. Below is the dynamic computer plot of the level surface for \(f(x,y,z) = \sin(3x)+y^2-z^2\). You can also explore further by changing the equation.
Nov 14, 2024 · Data Visualization. Calculus and Analysis. Differential Geometry of Surfaces. A level set in three dimensions.
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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.
