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Level curves of the function g(x,y)=√9−x2−y2 g (x y) = 9 − x 2 − y 2, using c=0,1,2 c = 0 1, 2, and 3 3 (c=3 c = 3 corresponds to the origin). A graph of the various level curves of a function is called a contour map.
The graph of a curve is given by $\{(a, f(a)): a \in \mathrm{dom}(f)\}$ where $f$ is a parametric representation of the curve. And the range of the curve is $\{f(a): a \in \mathrm{dom}(f)\}$. $\endgroup$
A level curve is simply a cross section of the graph of $z=f(x,y)$ taken at a constant value, say $z=c$. A function has many level curves, as one obtains a different level curve for each value of $c$ in the range of $f(x,y)$.
Contour Maps and Level Curves Level Curves: The level curves of a function f of two variables are the curves with equations where k is a constant in the RANGE of the function. A level curve is a curve in the domain of f along which the graph of f has height k. € f(x,y)=k € f(x,y)=k
Thus the graph of $z = f(x,\,y)$ can be visualized in two ways, one as a surface in $3$-space, the graph of $z = f(x,\,y)$, the other as a contour map in the $xy$-plane, the level curves of value $c$ for equally spaced values of $c$. As we shall see, both capture the properties of $z = f(x,\,y)$ from different but illuminating points of view.
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.
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A level curve is a type of level set. For $c=16$, the only point in the solution set is the origin, $x=y=0$. A single point is not a curve.
