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 TpS 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.