Functions of Several Variables
You can use level curves or contour curves to graph a function of 3 variables. Just set
z
=
C
, a
constant. Pick several values of C and visualize the curve from these level curves. If
z
=
C
intersects
z x
,
y
, the contour curve is just a trace.
Linear functions of three variables represent a plane. (Think the general plane equation.)
Limits and Continuity
concept behind
proofs.
lim
x
,
y
a
,
b
f x
,
y
=
L
iff for each
0 there is a
0 such that
f x
,
y
L
whenever
x
,
y
D and 0
x
x
0
2
y
y
0
2
.
x
x
0
2
y
y
0
2
defines a disc centered at
x
0
,
y
0
: since x and y are both variables, you can
approach a point infinitely many ways, from any direction on the xy plane (thus the circle). In two
dimensions, you could only approach the point two ways, from x or +x.
If you get different limits when you approach from different paths, the limit does not exist. This helps
us in finding continuous functions. If you can choose 2 different paths of approach to
x
0
,
y
0
and
they give you different limits, then the limit does not exist.
This is an example of a discontinuity in three dimensions.
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 Fall '07
 Bumpus
 Calculus, Derivative, Continuous function, tangency, ti ti ti

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