7.7 A Thick Parabola
The thick parabola is the last in our series of example surfaces.
This is the graph of the function
\[
\begin{align}
z &\depends (x, y)
\\
z &= x^2.
\end{align}
\]
Notice that one of the input variables \( y \) does not even appear in the definition of \( z \). We'll say that the variable \( y \) has been forgotten. We can draw this function as a schematic by using a forgetter.
A forgetter is a splitter that creates zero copies of its input. For more on forgetters, see Appendix C. Multi-Components. Let's take some vertical slices by fixing \( y \)-values.
\( \{ {y = 1} \} \)-slice.
We take the \( \{ {y = 1} \} \)-slice by substituting the number, one, for every occurrence of \( y \).
\[
\begin{align}
z &\depends x
\\
z &= x^2
\end{align}
\]
Since there are no occurrences of \( y \), no substitutions are needed.
\( \{ {y = -1} \} \)-slice.
Fixing a different \( y \)-value gives an identical looking slice.
\[
\begin{align}
z &\depends x
\\
z &= x^2
\end{align}
\]
Once again, there was no substitution needed because there is no dependence on \( y \).
These vertical slices help explain our surface: every \( (z \depends x) \)-slice of this function is just the standard parabola! Let's finish up by taking a quick look at the contour map for the thick parabola.
Each contour is a line. If we ever travel North or South on this contour map, we'll stay at the same height. To change our height, we should instead move East or West. Notice that the contours become more densely packed as we move away from the \( y \)-axis. Our surface is shallow near the \( y \)-axis and it becomes steeper moving to the East or West.