Surfaces, Contour Maps
John E. Gilbert, Heather Van Ligten, and Benni Goetz
The graph of a function
z
=
f
(
x, y
)
is the set of all points
{
(
x, y, f
(
x, y
)
}
. It’s a
surface
in
3
space that passes the
vertical line test
. But how can that surface be realized in
2
space, and how can
a particular realization help in understanding such important calculus features as slope, local extrema,
and concavity? One way, of course, is to ‘picture’ the surface just as Google does with ‘Street view’ or
a
45
◦
Earth Map view. Use Google Maps to see how this works near to where you live!
For example, the surface to the right is the graph
of
z
= sin(
x
) sin(
y
)
,

π
≤
x, y
≤
π
.
If di
ff
erentiation tells us about slope, how can we
express the slope at points on this surface, say at
the three dots or at the origin? Thinking of the
surface as mountains and valleys, the slope will be
positive when we are going uphill and negative
when we are going downhill. The locations of local
maxima and minima are clear from the picture, and
doesn’t the surface near the origin look like a
’Saddle’. What’s the slope at the origin? It
depends on the direction you are walking!
y
z
x
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But a Google map of mountains and valleys,
like the one to the right of part of Yosemite, also
represents the surface as a contour map with the
contours labelled by height. Recall that, for
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 Spring '07
 Sadler
 Calculus, Multivariable Calculus, Contour Map, Yosemite

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