COPYRIGHT NOTICE
Copyright 2007 Samuel Dagan
dagan@post.tau.ac.il
/general/copyright.xhtml
Constraints
Different constraints applyed on z=xy/2 are
discussed.
The origin of the figure is at (0,0)
©
Samuel Dagan
Constraints
x
y
-1.5
-1.0
-0.5
0.5
1.0
1.5
-1.5
-1
-0.5
0.5
1.0
1.5
0.25
0.25
0.5
0.5
1
1
1.5
1.5
-.25
-.25
-0.5
-0.5
-1
-1
-1.5
-1.5
0
π/3
2π/3
Some level lines of z = xy/2
are plotted and marked with
their z values.
The level lines
of z = 0 are
the (x,y) axes.
The unique stationary value
is
a saddle point at the origin.
Constraints will be applyed,
yielding different
results.
y = x yields z' = x²/2
with a
minimum
at x=y=0, as shown.
y = −x yields z' = −x²/2
with
a maximum at the origin.
y+x = 1 is off
the origin,
and yields z' = x(1−x)/2
with a maximum at x=y=½ .
The constraint r = 2sinφ
is a
circle centred at (x,y) = (0,1)
with radius of one unit,
and
φ
is limited by [0,π).
It yields z'=sin(2φ)sin²φ
with three stationary points:
a maximum at φ=π/3 ,
a minimum at φ=2π/3 ,
a stationary point
at φ=0(π) - the
origin.
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