C3 06.03.2023
martes, 28 de febrero de 2023 11:38
Index
The curve in space in which the plane z = c cuts a surface z = ƒ=(x,y) is made up of the points
that represent the function value ƒ(x, y) = c. It is called the contour curve ƒ(x, y) = c
The countour curve is the circle in the plane .
f
(
x
,
y
) = 100
− −
= 75
x
2
y
2
+ = 25
x
2
y
2
z
= 75
The level curve is the cricle in the xy-plane.
f
(
x
,
y
) = 100
− −
= 75
x
2
y
2
+ = 25
x
2
y
2
FIGURE 14.5 A plane z = c parallel to the xy-plane intersecting a surface
z = ƒsx, yd produces a contour curve.
Functions of Three Variables
DEFINITION Level Surface
The set of points (x, y, z) in space where a function of three independent variables has a constant value
ƒ(x, y, z) = c is called a level surface of ƒ.
EXAMPLE 5 Describing Level Surfaces of a Function of Three Variables
Describe the level surfaces of the function
f
(
x
,
y
,
z
)
=
+ +
x
2
y
2
z
2
‾ ‾‾‾‾‾‾‾‾‾‾
√
Solution The value of f is the distance from the origin to the point (x,y,z). Each level surface
, is a sphere of radius c centered at the origin. Figure14.7
shows a cutaway
view of three of these spheres. The level surface consists of the origin alone.
=
c
,
c
> 0
+ +
x
2
y
2
z
2
‾ ‾‾‾‾‾‾‾‾‾‾
√
= 0
+ +
x
2
y
2
z
2
‾ ‾‾‾‾‾‾‾‾‾‾
√
FIGURE 14.7
The level surfaces of
are concentric spheres (Example 5).
f
(
x
,
y
,
z
)
=
+ +
x
2
y
2
z
2
‾ ‾‾‾‾‾‾‾‾‾‾
√
14.2 Limits and Continuity in Higher Dimensions
Limits
DEFINITION Limit of a Function of Two Variables
We say that a function ƒ(x, y) approaches the limit L as (x, y) approaches sx0, y0d, and write
f
(
x
,
y
) =
L
lim
(
x
,
y
)
→
(
,
)
x
0
y
0
if, for every number , there exist a corresponding number such that for all in the
domin of f,
𝜖
> 0
𝛿
> 0
(
x
,
y
)
whenever
f
(
x
,
y
)
−
L
<
𝜖
∣
∣
∣
∣
0 < <
𝛿
+
(
x
−
)
x
0
2
(
y
−
)
y
0
2
‾ ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
√
THEOREM 1 Properties of Limits of Functions of Two Variables
The following rules hold if L, M, and k are real numbers and
f
(
x
,
y
) =
L
lim
(
x
,
y
)
→
(
,
)
x
0
y
0
and
g
(
x
,
y
) =
M
lim
(
x
,
y
)
→
(
,
)
x
0
y
0
1. Sum Rule-Difference Rule:
(
f
(
x
,
y
) ±
g
(
x
,
y
)) =
L
±
M
lim
(
x
,
y
)
→
(
,
)
x
0
y
0
2. Product Rule:
(
f
(
x
,
y
)
⋅
g
(
x
,
y
)) =
L
⋅
M
lim
(
x
,
y
)
→
(
,
)
x
0
y
0
3. Constant Multiple Rule:
(any number
k)
kf
(
x
,
y
) =
kL
lim
(
x
,
y
)
→
(
,
)
x
0
y
0
4. Quotient Rule:
=
M
≠
0
lim
(
x
,
y
)
→
(
,
)
x
0
y
0
f
(
x
,
y
)
g
(
x
,
y
)
L
M
5. Power Rule: If r and s are integers with no common factors, and , then
𝒔
≠
𝟎
=
lim
(
x
,
y
)
→
(
,
)
x
0
y
0
(
f
(
x
,
y
))
r
/
s
L
r
/
s
provided is a real number. (If s is even, we assume that .)
L
r
/
s
L
> 0
