7.3 Horizontal Slices
We just learned how to take vertical slices of a function \( z \depends (x, y) \) with two input variables: we fix either of the input variables. We'll take a horizontal slice of the graph by fixing the output variable. Given any constant \( z_0 : \Number \), the horizontal slice
\[
\begin{align}
&\{ {z = z_0} \}
\end{align}
\]
consists of all points on the graph that have height \( z_0 \).
We see horizontal slices in \( (x, y, z) \)-coordinate space by looking down from above.
Unlike vertical slices, we do not expect a horizontal slice to be the graph of a function! In general, we have no reason to think that \( y \) depends on \( x \) or vice versa.
Contour Maps
A contour map for a function \( z \depends (x, y) \) gives a top-down picture of the function's graph. The contour map consists of several horizontal slices all drawn in the same \( (x, y) \)-coordinate plane. We'll call each slice a contour, and we'll label each contour with its \( z \)-value height.
Contour maps are often used in cartography to display the elevation of a landscape. If we imagine ourselves hiking around on a surface, we can gain or lose elevation by hiking from one contour to another. To maintain our elevation, we may hike along a contour.
When we work in the \( (x, y) \)-coordinate plane, we should be careful how we describe movement in the \( y \)-direction. While it can be tempting to use the words "up" and "down" for the \( y \)-direction, it's better to reserve these words for the \( z \)-direction. If we keep our picture of space in mind, we could use the words "back" and "forward" for the \( y \)-direction. Cartography offers another solution: we can use cardinal directions for the \( (x, y) \)-coordinate plane.
