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MAT237Y1 Study Guide - Level Set, Wavelength-Dispersive X-Ray Spectroscopy, Compact Space


Department
Mathematics
Course Code
MAT237Y1
Professor
all

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1. (a) Verify whether or not the following statements are correct. No marks for guessing.
(i) [3 marks] If
S, then
}0:)0,{(\}11:),{( 22 >= xxyxRyx
. )}0,0{(\}11:),{( 22int <<= yxRyxS
NO. }0:)0,{(\}11:),{( 22int <<= xxyxRyxS
(ii) [4 marks] Every continuous function attains its absolute minimum value
RRSf 2
:
and its absolute maximum value on the set , where denotes the line segment in
i
i
LS U
=
=
1
i
L
2
R
from the origin to the point )0,0()
1
1,
1
2
i
i
( on the circular arc 2
1xy = .
NO. By the Extreme Value Theorem, continuous function is sure to attain its absolute
minimum value and its absolute maximum value on the set if S is compact. S
Consider the sequence of points =
k
xS
k
k
)
1
1,
1
2
(, lim so by the S
k
k
=
)1,0(x
Bolzano-Weierstrass S is not compact. Hence the statement is false.
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1.(c) [5 marks] Consider the area A of the parallelogram generated by the vectors
u. Use differentials to answer the following question:
)0,5,1(),0,3,2( == v
“To which non-zero component of the vectors u, v is the value of the area A
most sensitive?”
(That is a small change of that component causes the biggest change of the value of A)
HINT: Show first that 51
32
=
A
=
++=×=
2
22
51
32
00|||| vuA|
51
32
| = 51
32
since 013
51
32 >=
Consider the function yzxw
wz
yx
wzyxA ==),,,(. Then
dA . At the point (xdwydzzdywdx += )5,1,3,2
we get dA dwdzdydx 235
+
+
=
.
Hence a small change of dx causes the largest change in A, and consequently A is most
sensitive to the entry 2 of the vector u.
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