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Multivariable Functions, Surfaces, and Contours

The graphs of surfaces in 3-space can get very intricate and complex! In this tutorial, we investigate some tools that can be used to help visualize the graph of a function $f(x,y)$, defined as the graph of the equation $z=f(x,y)$.
Try plotting $$ z=\sin (xy)! $$

Example

Let $f(x,y) = x^2 + \frac{y^2}{4}$. 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?

View Surface

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.

Example

Let $f(x,y) = 5 - \sqrt{x^2 +y^2}$. 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 $z=5-x$ and $z=5+x$.

Similarly, the trace in the plane $x=0$ is the pair of lines $z=5-y$ and $z=5+y$. The surface is a right circular cone.

View Surface

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 $k$.

A collection of level curves of a surface, labeled with their heights, is called a contour map.
A contour map is just a topographic map of the surface.

Example

Let $f(x,y) = \sqrt{9-x^2-y^2}$. 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 $z=\sqrt{9-x^2-y^2}$ 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.

Note

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 $f(x,y)$.