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Math 454, Spring ’08, homework assignment #2, due Feb 14
Problem 1
The
Peano curve
is a continuous curve
r
(
t
) = (
x
(
t
)
, y
(
t
))
,
0
≤
t
≤
1
which passes through every point of the unit square 0
≤
x, y
≤
1. Explain why it
cannot be rectiﬁable. (Yes, such pathological curves do exist! Both Peano’s fourpage
paper in
Math. Ann
36, 1890 and Hilbert’s paper in
Math. Ann.
38, 1891, just under
two pages, are available on the web. You don’t need to know the explicit construction
to solve this problem.)
Problem 2
The Cauchy–Crofton formula for the arc length states that
2
L
=
ZZ
Ndpdθ.
Use it to ﬁnd the arc length of a quarter circle. (Apply the formula directly, do not
reduce to the case of the full circle.)
Problem 3
Dido’s problem
asks to ﬁnd among the arcs of given length
L
with end
points on a straight line the arc which together with the line segment between the
endpoints encloses the region of maximal area (draw a picture!).
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This homework help was uploaded on 02/24/2008 for the course MATH 4540 taught by Professor Protsak during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 PROTSAK
 Math, Geometry

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