Surfaces in R3

This chapter studies the curvature of a C² surface S ⊂ R³ at a point p ∊ S. (C² means that locally, the surface is the graph of a function just with continuous second derivatives. Computing the curvature will involve differentiating twice.) Let T_pS denote the tangent space of vectors tangent to S at p. Let n denote a unit normal to S at p. To study the curvature of S, we slice S by planes containing n and consider the curvature vector κ of the resulting curves. (See Figure 3.1.) Of course each such κ must be a multiple of n : κ = κn. (For now we will allow κ to be positive or negative. The sign of κ, depends on the choice of unit nonnal n.) It will turn out that the largest and the smallest curvatures κ₁, κ₂ (called the principal curvatures) occur in orthogonal directions and determine the curvatures in all other directions.