I'm starting to realize that calculus (or rather analysis) is pretty important for UI design. Programmers think in terms of discrete things but users think in terms of a continuous world (no sudden jumps between 'things'). However, user interfaces often consist of discrete elements. Yet since users think in terms of continuity they create a mental model that fills in the apparent gaps. So I'm beginning to appreciate that good UI's are the ones where arbitrarily small actions have similarly small effects (e.g. the scroll bar and certain animations) which allows for a smooth learning curve (I'm using the term 'smooth' in the same sense that we use it in calculus).

I think one of the reasons that OSX and other Apple products have a reputation of being so approachable is that they give the illusion of a continuous mapping between user actions and results.

(I just re-watched the video on seam-carving and even that seems like one of those obvious ideas once you have a good grasp of analysis.)