Cylinders, Quadric Surfaces, and Slicing
John E. Gilbert, Heather Van Ligten, and Benni Goetz
Now, fnally, we are almost ready to develop calculus oF Functions oF several variables. Let’s start by
trying to understand the graph oF some particular Functions
z
=
f
(
x, y
)
and, more generally, relations
f
(
x, y, z
) = 0
. Some examples oF relations
f
(
x, y, z
) = 0
that will later become crucial make a good
starting point.
Planes:
we’ve seen that a plane can be described by a
Linear Equation
Ax
+
By
+
Cz
=
D
where the vector
n
=
a
A, B, C
A
oF coe±cents is normal,
i.e.
, perpendicular, to the plane. Linear here
means that all the variables have power
one
. The graph oF
x
+
y
+
z
= 2
is shown in pink to the
right below. To see why, let’s start with the coordinate planes and axes as shown to the leFt below.
Recall that a plane is determined by three points on it or by any pair oF lines in the plane.
²or 3 points, take the intercepts
x
= 2
, y
= 2
,
and
z
= 2
on the coordinate axes. Do you see
them on the right?
What’s a natural pair oF lines in the pink plane to
use?
x
+
y
+
z
= 2
How about taking the intersection oF
x
+
y
+
z
= 2
with the
y
= 0
plane? This is the line
x
+
z
= 2
;
see it to the right? ²or the other line we could take the intersection
y
+
z
= 2
with the
x
= 0
plane.
This idea oF slicing, oFten by the coordinate planes, will be a crucial tool For us.