Unformatted text preview: 268 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION
is a diﬀerent parabola
z = 4x2 .
One might say that the “shadow” of the graph on the xz -plane is a
narrower parabola than the shadow on the yz -plane. (See Figure 3.35.)
This surface is called an elliptic paraboloid.
A more interesting example is given by the function
f (x, y ) = x2 − y 2 .
The horizontal slice at height c = 0 is a hyperbola which opens along the
x-axis if c > 0 and along the y -axis if c < 0; the level set L(f, 0) is the pair
of diagonal lines
y = ±x
which are the common asymptotes of each of these hyperbolas. (See
To see how these ﬁt together to form the graph, we again slice along the
coordinate planes. The intersection of the graph
z = x2 − y 2
with the xz -plane
is a parabola opening up : these points are the “vertices” of the hyperbolas
L(f, c) for positive c. The intersection with the yz -plane
is a parabola opening down, going through the vertices of the hyperbolas
L(f, c) for negative c. Fitting these pictures together, we obtain a surface shaped like a saddle
(imagine the horse’s head facing parallel to the x-axis, and the rider’s legs
parallel to the yz -plane). It is often called the saddle surface, but its
oﬃcial name is the hyperbolic paraboloid. (See Figure 3.7.) ...
View Full Document
This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08