Chapter 8. Partials
By studying derivatives we learned to calculate slope for functions that have one input variable. Partials are a generalization of derivatives that work for functions with more input variables. We'll soon see that partials are slopes that are found in a surface's vertical slices.
Moving to higher partials, we'll find that a surface has both bend and warp. We've studied bend with derivatives, and bend will generalize in a simple way. Warp, however, is something new for surfaces! The warp saddle gives an excellent illustration of warp.
Imagine holding a square of some flexible material on its left and right edges. We create a warped surface by twisting those edges in opposite directions, one clockwise and the other counterclockwise. Using partials, we'll be able to calculate a surface's warp at any point. In Chapter 9. Differentials as Metrics, we'll learn how to decide whether a surface is saddle-shaped. Warp will be an important, but not exclusive, factor in making this decision!