Try plotting z=sin(xy) !

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).

Example

Let f(x,y) = x² + (y²)/4. Before actually graphing z = x² + (y²)/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 = x².

Similarly, setting x = 0, the intersection of the surface with the yz-plane is the parabola z = (y²)/4.

Can you picture 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.

Example

Let f(x,y) = 5 - √(x² + y²). What can we determine about the surface given by z = 5 - √(x² + y²)? Notice that z ≤ 5. If we set z = 5, x² + y² = 0 and we get a single point x = 0, y = 0 in the plane z = 5.

If we set z = 4, then x²+y² = 1, giving a circle of radius 1.

If z = 0, then x² + y² = 5, a circle of radius √5.

If z = -4, then x² + y² = 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.

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 contour map is just a topographic map of the surface.

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

Example

Let f(x,y) = √(9-x²-y²). Notice here that f(x,y) ≥ 0. We will examine the level curves of z = f(x,y).

Setting z = k, k ≥ 0, squaring both sides of the equation and rearranging terms, we find that the level curves of z = f(x,y) are cirles given by x² + y² = 9 - k².

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 = √(9-x²-y²) from the previous example and rearranging terms, we obtain x² + y² + z² = 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.

In the following Exploration, choose a function f(x,y) to graph. You can view level cuves and build up a contour map for f(x,y).

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).