Lecture Notes on Surfaces

1. Surfaces Intuitively, a surface is a subset of R3 that “looks like” a plane when you zoom in on it, in the same way that the graph of a smooth function looks like a line when you zoom in on it. Surfaces are more complex objects than curves, and to appropriately describe them we will require a more complex definition than we had for curves. In this section we will gradually build the definition of a surface so as to understand the meaning of each of its requirements. We start with a definition that generalizes the notion of a curve.