7.4 A Bowl

In the remainder of this chapter we'll be looking at several examples of surfaces. Let's start by studying the bowl

which is the graph of the function

\[ \begin{align} z &\depends (x, y) \\ z &= x^2 + y^2. \end{align} \]

We'd like to understand this surface by taking some vertical slices and by looking at some contours.

\( \{ {y = 0} \} \)-slice.

Setting \( y \) equal to zero gives a vertical slice.

\[ \begin{align} z &\depends x \\ z &= x^2 \end{align} \]

This is a parabola in the \( (x, z) \)-plane. This slice can be seen in the surface by looking from the front.

\( \{ {x = 0} \} \)-slice.

Setting \( x \) equal to zero also gives a vertical slice.

\[ \begin{align} z &\depends y \\ z &= y^2 \end{align} \]

This is the same parabola, except it sits in the \( (y, z) \)-plane. This slice can be seen in the surface by looking from the right side.

The \( \{ {y = 0} \} \) and \( \{ {x = 0} \} \) vertical slices go a long way in helping us understand our surface. They do not, however, capture the fact that our bowl is round. We'll turn to horizontal slices, or contours, to see this aspect of our surface.

\( \{ {z = 1} \} \)-contour.

When we set our output variable, \( z \), equal to one

\[ \begin{align} &\bigl\{ {x^2 + y^2 = 1} \bigr\} \end{align} \]

we find the equation for a circle with radius one. We can locate this contour in our surface as the horizontal slice at height one. This slice should be visible in the surface by looking down from above.

\( \{ {z = 0} \} \)-contour.

Now let's set our output variable, \( z \), equal to zero.

\[ \begin{align} &\bigl\{ {x^2 + y^2 = 0} \bigr\} \end{align} \]

To satisfy this equation both \( x \) and \( y \) must be zero. The only point \( (x, y) \) that satisfies this equation is \( (0, 0) \). In other words, the horizontal slice at height zero only intersects the bowl at one point, the origin.

\( \{ {z = -1} \} \)-contour.

Setting \( z \) equal to negative one gives an equation that cannot be satisfied!

\[ \begin{align} &\bigl\{ {x^2 + y^2 = -1} \bigr\} \end{align} \]

Both \( x^2 \) and \( y^2 \) must be at least zero, and so their sum cannot be negative. What does this say about our graph? If we attempt to slice our surface at height negative one, we won't intersect the bowl.

Great! These contours tell us a good deal of information about the shape of the bowl beyond the vertical slices. Let's wrap up our analysis of the bowl by looking at a contour map.

Our \( \{ {z = 0} \} \) and \( \{ {z = 1} \} \) contours can be seen on this map. We've gone ahead and added a few more contours to give a better sense of the bowl. Notice how the spacing between contour lines varies across the contour map. The bowl is a shallow surface near the origin, but it becomes steeper as we move away from the origin. A shallow graph has sparsely packed contours, and a steep graph has densely packed contours.