COPYRIGHT NOTICE
Copyright 2007 Samuel Dagan
dagan@post.tau.ac.il
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Level lines (1)
Level lines of a function of two variables
The origin of the figure is at (0,0)
©
Samuel Dagan
Level lines (1)
x
y
-3
-2
-1
1
2
3
-3
-2
-1
1
2
3
0
1
2
3
The domain of the function
z = √
(9−x²−y²) is
x²+y²
≤ 9.
See some level lines next.
z = 0 yields
x²+y² = 9 .
z = 1 yields
x²+y² = 8 .
z = 2 yields
x²+y² = 5 .
z = 3 yields
x²+y² = 0 .
By taking into account that
the z axis is
towards us
(perpendicular to the plane),
and the consecutive level
lines are equally spaced on
z , one can
observe that:
The level lines are circles.
The steep rise of z at z=0
becomes more and more
moderate toward its peak.
The intersection of the
function with the (x,z) plane
(y = 0) ,
yields for z ≥ 0
z =
√(9−x²)
or x²+z² = 9 .
In view of all those, the
the function is half of a
spherical shell with
radius = 3.
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