400
CHAPTER 3. REALVALUED 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
i
0
)
2
(if
λ
i
n
= 0) or
α
i
(
ξ
i
−
ξ
i
0
) (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
shapeclassiFcation 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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 ALL
 Calculus, Completing The Square, Quadratic equation, Conic section

Click to edit the document details