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Solutions Manual
Elasticity: Theory, Applications and Numerics Second Edition
By
Martin H. Sadd
Professor Department of Mechanical Engineering & Applied
Mechanics
University of Rhode Island Kingston, Rhode Island
Foreword
Exercises found at the end of each chapter are an important
ingredient of the text as they provide homework for student
engagement, problems for examinations, and can be used in class to
illustrate other features of the subject matter. This solutions
manual is intended to aid the instructors in their own particular
use of the exercises. Review of the solutions should help determine
which problems would best serve the goals of homework, exams or be
used in class. The author is committed to continual improvement of
engineering education and welcomes feedback from users of the text
and solutions manual. Please feel free to send comments concerning
suggested improvements or corrections to [email protected]. Such
feedback will be shared with the text user community via the
publisher’s web site. Martin H. Sadd January 2009
Copyright © 2009, Elsevier Inc. All rights reserved.
mailto:[email protected]�
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Full file at https://fratstock.eu
1-1.
(scalar)5401
(matrix)402000201
(scalar)7400000201
(vector)243
(matrix)350
10180461
110240111
110240111
(scalar)251104160111
(scalar)6141 (a)
332211
332313
322212
312111
333323321331322322221221311321121111
332211
333332323131232322222121131312121111
332211
=++=++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=++++++++=
++++++++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=++++++++=
++++++++==++=++=
bbbbbbbbbbbbbbbbbbbbbbbbbb
bb
bbabbabbabbabbabbabbabbabbabba
babababa
aa
aaaaaaaaaaaaaaaaaaaaaaaa
ii
ji
jiij
iiijij
jkij
ijij
ii
(scalar)6114
(matrix)112112224(scalar)17240120044
(vector)634
(matrix)8160480261
240120021
240120021
(scalar)304160140041
(scalar)5221 (b)
332211
332313
322212
312111
333323321331322322221221311321121111
332211
333332323131232322222121131312121111
332211
=++=++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=++++++++=
++++++++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=++++++++=
++++++++==++=++=
bbbbbbbbbbbbbbbbbbbbbbbbbb
bb
bbabbabbabbabbabbabbabbabbabba
babababa
aa
aaaaaaaaaaaaaaaaaaaaaaaa
ii
ji
jiij
iiijij
jkij
ijij
ii
Copyright © 2009, Elsevier Inc. All rights reserved.
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Full file at https://fratstock.eu
(scalar)2011
(matrix)000011011
(scalar)3000001011
(vector)112
(matrix)1841931722
410201111
410201111
(scalar)251610401111
(scalar)5401 (c)
332211
332313
322212
312111
333323321331322322221221311321121111
332211
333332323131232322222121131312121111
332211
=++=++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=++++++++=
++++++++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=++=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=++++++++=
++++++++==++=++=
bbbbbbbbbbbbbbbbbbbbbbbbbb
bb
bbabbabbabbabbabbabbabbabbabba
babababa
aa
aaaaaaaaaaaaaaaaaaaaaaaa
ii
ji
jiij
iiijij
jkij
ijij
ii
1-2.
conditions eappropriat esatisfy th andclearly011101110
21
231381112
21
)(21)(
21)a(
][)( ijij
jiijjiijij
aa
aaaaa
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
−++=
conditions eappropriat esatisfy th andclearly030302
020
21
450542022
21
)(21)(
21)b(
][)( ijij
jiijjiijij
aa
aaaaa
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
−++=
Copyright © 2009, Elsevier Inc. All rights reserved.
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Full file at https://fratstock.eu
conditions eappropriat esatisfy th andclearly011100100
21
831302122
21
)(21)(
21)c(
][)( ijij
jiijjiijij
aa
aaaaa
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
−++=
1-3.
0011100100
831302122
41 :2(c)-1 Exercise From
0030302
020
450542022
41 :2(b)-1 Exercise From
0011101110
231381112
41 :2(a)-1 Exercise From
002
][)(
][)(
][)(
=⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
=⇒=⇒−=−=
T
ijij
T
ijij
T
ijij
ijijijijijijjijiijij
traa
traa
traa
bababababa
1-4.
ij
jkij
iiiijij
aaaaaaaaaa
aaaaaaaaaa
aaaa
aaaaaaaaa
aaaa
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⋅⋅⋅⋅⋅⋅
δ+δ+δδ+δ+δδ+δ+δ=δ
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
δ+δ+δδ+δ+δδ+δ+δ
=δ+δ+δ=δ
333231
232221
131211
331323121311321322121211311321121111
3
2
1
333232131
323222121
313212111
332211
Copyright © 2009, Elsevier Inc. All rights reserved.
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Full file at https://fratstock.eu
1-5.
333231
232221
131211
312232211331233321123223332211
332112322311312213322113312312332211
332112213322311132312213321
322113312312312231332211123321
)()()(
)det(
aaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaa
aaaaaaaaaaaaa kjiijkij
=
−+−−−=−−−++=
ε+ε+ε+
ε+ε+ε=ε=
1-6.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−==′
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−==′
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−==′
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−==′
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−==′
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−==′
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=⇒
5.05.12/25.25.32/2
021
2/22/202/22/20
001
410201111
2/22/202/22/20
001
2/22/2
1
011
2/22/202/22/20
001 :1(c)-1 Exercise From
5.05.105.15.40221
2/22/202/22/20
001
240120021
2/22/202/22/20
001
02
2
112
2/22/202/22/20
001 :1(b)-1 Exercise From
120140
021
2/22/202/22/20
001
110240111
2/22/202/22/20
001
22
1
201
2/22/202/22/20
001 :1(a)-1 Exercise From
2/22/202/22/20
001 axis-about xrotation 45 1
o
T
pqjqipij
jiji
T
pqjqipij
jiji
T
pqjqipij
jiji
ij
aQQa
bQb
aQQa
bQb
aQQa
bQb
Q
Copyright © 2009, Elsevier Inc. All rights reserved.
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Full file at https://fratstock.eu
1-7.
⎥⎦
⎤⎢⎣
⎡
θ+θθ+−θθ−θθ−−θθ−θθ−−θθ+θθ++θ
=
⎥⎦
⎤⎢⎣
⎡θθ−θθ
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡θθ−θθ
==′
⎥⎦
⎤⎢⎣
⎡θ+θ−θ+θ
=⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡θθ−θθ
==′
⎥⎦
⎤⎢⎣
⎡θθ−θθ
=⎥⎦
⎤⎢⎣
⎡
θθ+θ−θ
=⎥⎦
⎤⎢⎣
⎡′′′′
=
2222112
211
2122211
221
2212211
212
2222112
211
2221
1211
21
21
2
1
2212
2111
coscossin)(sinsincossin)(cossincossin)(cossincossin)(cos
cossinsincos
cossinsincos
cossinsincos
cossinsincos
cossinsincos
cos)90cos()90cos(cos
),cos(),cos(),cos(),cos(
aaaaaaaaaaaaaaaa
aaaa
aQQa
bbbb
bb
bQb
xxxxxxxx
Q
T
pqjqipij
jiji
o
o
ij
1-8.
ijjpippqjqipij aQaQaQQa δ==δ=δ′' 1-9.
jkiljlikklijlmknjnimkmjnimlpkpjmim
npmqnqmppqmnlqkpjnimjkiljlikklij
QQQQQQQQQQQQ
QQQQ
δγδ+δβδ+δαδ=γ+β+α=
δγδ+δβδ+δαδ=δ′δ′γ+δ′δ′β+δ′δ′α
ln
)('''
1-10.
klijlikjljkiijkl
jkiljlikklijjkiljlikklijijkl
C
C
=δδ+δδβ+δαδ=
δδ+δδβ+δαδ=δγδ+δβδ+δαδ=
)(
)(
1-11.
321
3
2
1
3132213
1
3
2
2
1
321
3
2
1
000000
00
00
00
000000
If
λλλ=λ
λλ
=
λλ+λλ+λλ=λ
λ+
λλ
+λ
λ=
λ+λ+λ==
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
λλ
λ=
a
a
iia
III
II
aI
a
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1-12.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −==′
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −=
±=⇒===⇒
=++=−
=+−⇒=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−
=λ
±=⇒±==⇒
=++=
=+−⇒=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−
=λ
−±=±=−=⇒=++
==+
⇒=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=λ=λ=λ−=λ⇒
=−λ+λλ⇒=−λ+λλ⇒=λ+λ−λ−∴
=−=−=⇒⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−=
100000002
2/200011011
100011011
2/200011011
21
and2/200
011011
2/2by given ismatrix rotation The
)1,0,0(1,01
0202
0000021012
:Case1
)011)(2/2(2/2
10
00
100011011
:Case0
)0,1,1)(2/2(,2/21
00
0300011011
:Case21,0,2 Roots
0)1)(2(0)2( 02 isEqn sticCharacteri
0,2,1100011011
(a)
)3()3(321
2)3(3
2)3(2
2)3(1
)3(2
)3(1
)3(2
)3(1
3
2
1
3
)2(21
2)2(3
2)2(2
2)2(1
)2(3
)2(2
)2(1
3
2
1
2
)1()1(2
)1(1
2)1(3
2)1(2
2)1(1
)1(3
)1(2
)1(1
)1(3
)1(2
)1(1
1
321
223
T
pqjpipij
ij
aaaij
aQQa
Q
nnnnnn
nnnn
nnn
,,nn
nnnn
nn
nnn
nnnnn
nnn
nnn
IIIIIIa
n
n
n
Copyright © 2009, Elsevier Inc. All rights reserved.
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Full file at https://fratstock.eu
1-12.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −==′
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −=
±=⇒===⇒
=++=−
=+−⇒=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−
=λ
±=⇒±==⇒
=++=
=+−⇒=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−
−=λ
−±=±=−=⇒=++
==+
⇒=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=λ=λ−=λ−=λ⇒
=+λ+λλ⇒=+λ+λλ⇒=λ−λ−λ−∴
==−=⇒⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−=
000010003
2/200011011
000021012
2/200011011
21
and2/200
011011
2/2by given ismatrix rotation The
)1,0,0(1,01
0202
0000021012
:Case0
)011)(2/2(2/2
10
00
100011011
:Case1
)0,1,1)(2/2(,2/21
00
0300011011
:Case30,1,3 Roots
0)1)(3(0)34( 034 isEqn sticCharacteri
0,3,4000021012
(b)
)3()3(321
2)3(3
2)3(2
2)3(1
)3(2
)3(1
)3(2
)3(1
3
2
1
3
)2(21
2)2(3
2)2(2
2)2(1
)2(3
)2(2
)2(1
3
2
1
2
)1()1(2
)1(1
2)1(3
2)1(2
2)1(1
)1(3
)1(2
)1(1
)1(3
)1(2
)1(1
1
321
223
T
pqjpipij
ij
aaaij
aQQa
Q
nnnnnn
nnnn
nnn
,,nn
nnnn
nn
nnn
nnnnn
nnn
nnn
IIIIIIa
n
n
n
Copyright © 2009, Elsevier Inc. All rights reserved.
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Full file at https://fratstock.eu
1-12.
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −==′
⎥⎥⎥
⎦
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⎢⎢⎢
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⎡ −=
±=±==
±=⇒−==⇒=++
=+−⇒=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−
=λ=λ
−±=±=−=⇒=++
==+
⇒=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
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−=λ=λ=λ−=λ⇒
=+λλ=λ−λ−∴
==−=⇒⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−=
000000002
2/200011011
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2/2by given ismatrix rotation The
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choosemay weeconveniencFor .determineduniquely not are
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1-13*.
Copyright © 2009, Elsevier Inc. All rights reserved.
-
Full file at https://fratstock.eu
1-14.
3
1
23
2
2
2122132
22
321
33,32,21,1
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34)(,300
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322
2 (b)
21)(,222
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212 (a)
xtrx
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xuuuxxxx
xxtrx
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x
xxxxx
xxx
xxxuuu
xxxx
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xx
xxxxxxxxxxx
xxx
xxxuuuxxxxxx
=∇⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=∇
=++=∇
−−−=∂∂∂∂∂∂=×∇
++=++=⋅∇++=
+=∇⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=∇
=++=∇
+−=∂∂∂∂∂∂=×∇
++=++=⋅∇
++=
++=∇⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=∇
=++=∇
+−=∂∂∂∂∂∂=×∇
++=++=⋅∇++=
uu
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u
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u
ueeeu
321
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Copyright © 2009, Elsevier Inc. All rights reserved.
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Full file at https://fratstock.eu
1-15.
iijkjk
mnmnmnnmmn
jkkmjnknjmjk
knkmki
jnjmji
inimii
jkimnijkiimn
jkijki
aa
aaaaa
aaaa
aa
ε−=∴
−=+−=−−=
δδ−δδ−=δδδδδδδδδ
−=εε−=ε
ε−=
)(21)(
21
)(21
21
21
21
1-16.
)()()(2)()(
2)()()(
)()( (a)
,,,
22
,,,,,,,,,,,,,,2
,,,
uuu ⋅∇φ+⋅φ∇=φ+φ=φ=φ⋅∇ψ⋅φ+ψ∇φ+ψφ∇=
ψφ+φψ+ψφ=ψφ+ψφ+ψφ+φψ=ψφ+φψ=φψ=φψ∇
ψ∇φ+φψ∇=ψφ+φψ=φψ=φψ∇
kkkkkk
kkkkkkkkkkkkkkkkkkk
kkk
uuu∇∇
φ∇=φ=φ=φ⋅
=φε=φε=φ×
×⋅−×⋅=ε+ε=+ε=ε=×⋅
×φ+×φ=φε+φε=φ+φε=φε=φ×
2,,,
,,,
,,,,,
,,,,
)(
inryantisymmetandsymmetryofbecause0)(
)()()()()
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kijkkjijkkjjkijkjkijk
jk
vuuvvuvuvu
uuuuu
∇∇
∇∇
∇∇(∇
∇∇((∇
vuuvvu
uuu
uuuu
uuuu
u
u
∇⋅−⋅∇=
−=δδ−δδ=εε=εε=××∇−⋅=
−=δδ−δδ=εε=εε=××
=ε=ε=×⋅
)(21
)()()()(
)()()(in ry antisymmet andsymmetry of because,0)()(
(c)
,,,,,
2
,,,,,,
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nnmnmnjnknjmknkmjjnkijkimnnjkijkmni
jikijkijkijk
uuuuuuuuuu
uuuuuijuu
∇∇∇
∇∇
∇∇
Copyright © 2009, Elsevier Inc. All rights reserved.
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Full file at https://fratstock.eu
1-17.
zθr
zθr
zθr
zzzθrr
θθ
r
3z21θ21r
eeeu
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2
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22
3212222
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⎟⎠⎞
⎜⎝⎛
θ∂∂
−∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
−∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
−θ∂
∂=×
∂∂
+θ∂
∂+⎟
⎠⎞
⎜⎝⎛
∂∂
∂∂
=∇
∂∂
+θ∂
∂+
∂∂
=⋅
∂∂
+θ∂∂
+∂∂
=
∂∂
+θ∂∂
+∂∂
=
=∂∂
=θ∂
∂=
∂∂
=∂∂
=∂∂
−=θ∂
∂=
θ∂∂
=θ+θ−=θ+θ====⇒+θ+=
=ξθ=ξ=ξ
θθ
θ
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zuu
r
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rrfr
rrf
zuu
rru
rr
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rrff
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hrhhdzrddrdszr
∇
∇
∇
∇
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Full file at https://fratstock.eu
1-18.
φ=⇒φ=ξ∂∂
ξ∂∂
=
=⇒=ξ∂∂
ξ∂∂
=
=⇒=φ+θφ+θφ=ξ∂∂
ξ∂∂
=
ξξ=ξξξ=ξξξ=
θ=ξφ=ξ=ξ
sinsin)(
)(
11cos)sin(sin)cos(sin)(
:factorsScalecos,sinsin,cossin
,,:scoordinateSpherical
322
332
3
22
222
2
1222
112
1
21332123211
321
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RhRxxh
hxxh
xxxR
kk
kk
kk
φθθθ
φφφ
φ
φ
φ−=θ∂
∂=
φ∂∂
=∂∂
φ=θ∂
∂−=
φ∂∂
=∂∂
φ=θ∂
∂=
φ∂∂
=∂∂
θ+θ−=
φ−φθ+φθ=φ+φθ+φθ=
eeee
ee
eee
eeeeeeee
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θ
θRRR
21θ
321
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ˆcosˆ
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:vectorsUnit
R
R
R
r
2
2
2222
2
222
22
22
222
2
sin1)(sin
sin11
)sin
1(sin1)(sin
sin1sin
sin1
)(sin1)(sin
sin1)(1
)(sin1)sin(
sin1)sin(
sin1
sin1ˆ1ˆˆ
sin1ˆ1ˆˆ
(1.9.16) - (1.9.12) Using
θ∂∂
φ+
φ∂∂
φφ∂∂
φ+⎟
⎠⎞
⎜⎝⎛
∂∂
∂∂
=
θ∂∂
φθ∂∂
φ+
φ∂∂
φφ∂∂
φ+⎟
⎠⎞
⎜⎝⎛
∂∂
φ∂∂
φ=∇
θ∂∂
φ+φ
φ∂∂
φ+
∂∂
=
θ∂∂
φ+φ
φ∂∂
φ+φ
∂∂
φ=⋅
θ∂∂
φ+
φ∂∂
+∂∂
=
θ∂∂
φ+
φ∂∂
+∂∂
=
⇒
θφ
θφ
φ
θφ
fR
fRR
fRRR
fR
fRR
fRRR
f
uR
uR
uRRR
RuR
uRR
uRRR
fR
fRR
ff
RRR
R
R
R
R
u
eee
eee
z
∇
∇
∇
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Full file at https://fratstock.eu
1-18. Continued
θ
φR
e
ee
e
eeu
ˆ)(1
ˆ)(1sin1ˆ)(sin
sin1
ˆ)()[(1
ˆ)]sin()([sin1ˆ)]()sin([
sin1
2
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛φ∂
∂−
∂∂
+
⎥⎦
⎤⎢⎣
⎡∂∂
−θ∂
∂φ
+⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛θ∂
∂−φ
φ∂∂
φ=
⎟⎟⎠
⎞⎜⎜⎝
⎛φ∂∂
−∂∂
+
⎟⎟⎠
⎞⎜⎜⎝
⎛φ
∂∂
−θ∂∂
φ+⎟⎟
⎠
⎞⎜⎜⎝
⎛θ∂∂
−φφ∂∂
φ=×
φ
θφ
θ
θφ
φθφθ
R
R
R
RR
uRuRR
RuRR
uR
uu
R
uRuRR
uRR
uR
RuuRR
∇
Copyright © 2009, Elsevier Inc. All rights reserved.
-
Full file at https://fratstock.eu
2-1.
−−
−−
−−
=−=ω
⋅⋅⋅
++
=+=
+===
−
−−
=−=ω
⋅⋅
⋅=+=
===
+−
−−−
−−
=−=ω
⋅⋅+⋅
+
=+=
+===
00)23(21
00))(21
)23(21)(
210
)(21
202
)23(21)(
210
)(21
)(,, (c)
021
21
210
21
21
210
)(21
2121
212
)(21
,, (b)
0
0)(21
)(210
)(21
00
)(21
)(21
)(,, (a)
2
23
223
,,
223
,,
2223
,,
,,
2
2
2
,,
2
,,
222
CxAyz
ByAz
CxAyzByAz
uu
CzBxy
CxAyzByAz
uue
zxCwBxyvAyzu
CxzCyz
CxzBy
CyzBy
uu
Cxy
CxzBx
CyzByAx
uue
CxyzwBxyvAxu
CyBxzCx
CyBxzBzAx
CxBzAx
uu
CyBxz
CxBzAxAy
uue
yxCwBxzvAxyu
ijjiij
ijjiij
ijjiij
ijjiij
ijjiij
ijjiij
Copyright © 2009, Elsevier Inc. All rights reserved.
Solutions Manual Foreword
2eCH01Solutions2eCH02Solutions2eCH03Solutions2eCH04Solutions2eCH05
Solutions2eP
CH06 Solutions2eCH07 Solutions2eCH08 Solutions2eBA(pL
CH09 Solutions2eP
CH10 Solutions2eCH11 Solutions2eCH12 Solutions2eCH13
Solutions2eCH14 Solutions2eCH15 Solutions2eCH16
Solutions2eCH03Solutions2eCH04Solutions2eCH05 Solutions2eP
CH06 Solutions2eCH07 Solutions2eCH08 Solutions2eBA(pL
CH09 Solutions2eP
CH10 Solutions2eCH11 Solutions2eCH12 Solutions2eCH13
Solutions2eCH14 Solutions2eCH15 Solutions2eCH16 Solutions2e
