
ExampleLet $f(x,y) = x^2 + \frac{y^2}{4}$. Before actually graphing $z = x^2 + \frac{y^2}{4}$, let's see if we can visualize the surface that will result. If we set $y=0$, we find that the intersection of the surface with the $xz$plane is the parabola $z=x^2$. Similarly, setting $x=0$, the intersection of the surface with the $yz$plane is the parabola $z=\frac{y^2}{4}$. Can you see what the surface, called an elliptic paraboloid, will look like?
By setting $x=0$ or $y=0$ in $z=f(x,y)$, we are really looking at the intersection of the surface $z=f(x,y)$ with the plane $x=0$ or $y=0$, respectively. If we take the intersection of a surface $z=f(x,y)$ with any plane, the resulting curve is called the cross section or trace of the surface in the plane.
ExampleLet $f(x,y) = 5  \sqrt{x^2 +y^2}$. What can we determine about the surface given by $z = 5  \sqrt{x^2 +y^2}$? Notice that $z \leq 5$. If we set $z=5$, $x^2 + y^2 =0$ and we get a single point $x=0, \quad y=0$ in the plane $z=5$. If we set $z=4$, $x^2+y^2 =1$, giving a circle of radius 1. If $z=0$, $x^2 + y^2 =5$, a circle of radius $\sqrt{5}$. If $z=4$, $x^2 + y^2 = 9$, a circle of radius 3. Is this another paraboloid? Notice that the trace in the plane $y=0$ is the pair of lines $z=5x$ and $z=5+x$. Similarly, the trace in the plane $x=0$ is the pair of lines $z=5y$ and $z=5+y$. The surface is a right circular cone.
When we take the intersection of the surface $z=f(x,y)$ with the horizontal plane $z=k$, as we did several times in the previous example, the projection of the resulting curve onto the $xy$plane is called the level curve of height $k$. Along this curve, $f$ is constant with value $k$.
ExampleLet $f(x,y) = \sqrt{9x^2y^2}$. Notice here that $f(x,y) \ge 0$. We will examine the level curves of $z=f(x,y)$. Setting $z=k, \quad k \ge 0$, squaring both sides of the equation and rearranging terms, we find that the level curves of $z=f(x,y)$ are circles given by $x^2 + y^2 = 9  k^2$. Examination of traces with $x=c$ or $y=c$ shows them to be portions of circles. Thus, $z=f(x,y)$ is a hemisphere here. Squaring $z=\sqrt{9x^2y^2}$ from the previous example and rearranging terms, we obtain $x^2 + y^2 + z^2 = 9$, the equation of a sphere. It is useful to be able to recognize some common quadric surfaces such as this. For a function $f(x,y,z)$ of three variables, $f(x,y,z) = k$ is called the level surface with constant $k$. The function $f(x,y,z)$ is constant over the level surface.
Key Concept
Let $z=f(x,y)$. The projection onto the $xy$plane of the intersection of the surface $z=f(x,y)$ with the horizontal plane $z=k$ is called the level curve of height $k$. A collection of level curves, called a contour map is a useful tool in visualizing the graph of a function $f(x,y)$. 