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# MAT237Y1 Study Guide - Piecewise, Density

Department
Mathematics
Course Code
MAT237Y1
Professor
all

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1. Suppose where a is a constant. kjiF yxazzyx ++=),,(
(a) [5 marks] Is there a real number a such that 0])[( >
Ć
FFcurldiv . Justify.
We have
)1,,1( a
yxaz zyx
curl =
ā
ā
ā
ā
ā
ā
=
kji
F and
=
Ć
FF)(curl ),,(11 2zaxyazxay
yxaz
aāāā=
kji
Hence for all values of a and so there is no value of a 011])[( 2<āāā=Ć acurldiv FF
such that . 0])[( >Ć FFcurldiv
(b) [7 marks] In case , evaluate , where C is the portion of the curve of
2ā=aā«ā
C
dxF
intersection of surfaces and in the first octant from o
2
xz =4
22 =+ yx )4,0,2( t )0,2,0(.
In this case kjiF yxzzyx
+
+ā= 2),,(. From and (both equations should
be satisfied) we get a required parametrization of C :
222 2=+ yx
x2
2
xz =
tcos
=
, tsiny2
=
, , tz 2
cos4=
2
0
Ļ
ā¤ā¤ t and we have
ā«ā
C
dxF = =
ā«āāāā
2/
0
2)sincos8,cos2,sin2()sin2,cos2,cos8(
Ļ
dtttttttt
= =
ā«ā+
2/
0
222 )cossin16cos4sincos16(
Ļ
dtttttt
=
ĻĻ
Ļ
=ā++āāā++=ā++ā )000
3
16
()
3
16
00(]sin
3
16
2sin2cos
3
16
[2/
0
33 tttt
2