400 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION by completing the square in each variable ξ i for which λ i n = 0 we get an equation in which each variable appears either in the form λ i ( ξ i − ξ i0 ) 2 (if λ i n = 0) or α i ( ξ i − ξ i0 ) (if λ i = 0). We shall not attempt an exhaustive catalogue of the possible cases, but will consider Fve “model equations” which cover all the important possibilities. In all of these, we will assume that ξ 10 = ξ 20 = ξ 30 = 0 (which amounts to displacing the origin); in many cases we will also assume that the coe±cient of each term is ± 1. The latter amounts to changing the scale of each coordinate, but not the general shape-classiFcation of the surface. 1. The easiest scenario to analyze is when z appears only to the frst power : we can then move everything except the “ z ” term to the right side of the equation, and divide by the coe±cient of z , to write our equation as the expression for the graph of a function of
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