Math 4250/6250 Homework #3
This homework assignment covers our notes on integral geometry (5) and on rotation index (6).
Please pick 3 of the following problems. Remember that undergraduate students should average
one
challenge problem per assignment, while graduate students should average
two
challenge
problems per assignment.
1. R
EGULAR
P
ROBLEMS
Do Carmo, p 47. #8, #12 (
There is a misprint in this problem! The ratio
M
1
/M
2
= 1
/
2
, not
1
/
3
as stated in the text
), #13.
In #12, DoCarmo asks for the “measure” of a set of lines. What is measure? We established
a system of coordinates for the set of lines when we proved the CauchyCrofton formula:
(
p, θ
)
is the line at distance
p
from the origin which is normal to angle
θ
. In these coordinates, we can
compute the “area” of a set of lines by integrating the function
1
over the set with respect to
p
and
θ
. So the “measure” or area of a set
S
of lines is just
Measure(
S
) =
S
1 d
p
d
θ.
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 Spring '08
 Staff
 Math, Geometry, total curvature

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