Chapter 7. Surfaces
It's time for us to improve our graphing game! Up to this point, we've only needed two axes for graphing:
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a vertical \( z \)-axis for output, and
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a horizontal \( x \)-axis for input.
This is enough when we look at functions \( z \depends x \) that only have one input variable. But we'd like to start graphing functions \( z \depends (x, y) \) with two input variables, and for this, we'll need another axis:
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a horizontal \( y \)-axis for input.
Because two of our three axes are for input variables, a graph will be two-dimensional, a surface, sitting in three-dimensional space!
It's our job, then, to learn how to visualize surfaces. But surfaces can be quite complicated to draw on paper or a screen. Even with years experience, many of our drawings won't quite capture their subject. So we offer a compromise: we'll study surfaces by slicing them. Slices are relatively easy to draw because they ignore some of the surface's complexity. By looking at multiple slices, we can go a long way towards reconstructing a surface!