I assume sharp vertices do appear in graphics. Are gradients still used in those situations, and if so, how?
@cou Beautiful question. The answer of course depends on what exactly you're trying to do, but if the goal is to treat a mesh with "sharp vertices" as a smooth surface, then there are a number of ideas. One standard approach is to fit a smooth function (e.g., a "spline") to this data, then do computations on this smooth function. Another is to re-interpret the "sharp" data as actually a proxy for a smooth object, i.e., re-think your definitions so that things just fit together naturally. This is the perspective of discrete differential geometry, and I teach a course on it in the Spring: Discrete Differential Geometry.
I used to think for directional derivative u should be a unit vector, but later found in assignment0.5 that u could have any magnitude. So what's the geographical meaning for u's magnitude?