Answers to Selected Problems: Part II, Chapter 3...
1
A3 Answers to Selected Problems: Part II, Chapter 3 3.1 1. ( ) ( ) ( ) ( ) ( ) ( ) 0 1 1 2 1 2 0 1 1 2 1 2 2 2 2 2 2 2 2 2 2 2 2 2 = + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ − + ∂ ∂ ∂ + + ∂ ∂ − = + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∂ ∂ − + ∂ ∂ ∂ + + ∂ ∂ − y y x y x x y x b x u y x u y u E b y u y x u x u E ν ν ν ν ν ν 2. ( ) ( ) ( ) ( ) y AE x AE x AE xy xx 2 2 2 1 2 , 2 1 2 , 1 2 ν σ ν ν ν σ + − = + + = + = 3. (a) Yes, (b) Yes, (c) y x u u , non-zero along the base ( xx ε is non-zero which in itself is inconsistent with 0 = x u along base) 3.2 3. B A 5 = , ( ) ( ) [ ] 3 2 10 30 2 1 x xy E B V V − − = Δ ν 4. [ ] [ ] [ ] 2 3 4 5 3 2 4 5 3 2 4 5 24 4 , 11 3 , 17 15 xy x y y x y y x E xy E yy E xx − − = + = − = ε ε ε [ ] [ ] A Cx y y x x u B Cy x y y x u E y E x + + + + − = + + − = 4 4 11 2 2 2 3 4 4 13 4 5 3 3 4 5 , 17 5 6. ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + + + − = 2 3 2 2 2 3 2 2 2 2 2 3 2 2 2 3 3 4 3 1 6 2 3 4 b L x b L b x b xy Eb F u y b L y b y b y x Eb F u y x ν ν ν 8. x g y x g gy xy yy xx β ρ σ β β ρ σ ρ σ 2 2 tan , tan 2 tan , − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = − =
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Transcript of Answers to Selected Problems: Part II, Chapter 3...
A3
Answers to Selected Problems: Part II, Chapter 3 3.1
1. ( ) ( ) ( )
( ) ( ) ( ) 011212
011212
2
22
2
2
2
2
22
2
2
2
=+⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂
∂−+
∂∂∂
++∂
∂
−
=+⎥⎥⎦
⎤
⎢⎢⎣
⎡
∂∂
−+∂∂
∂++
∂∂
−
yyxy
xxyx
bxu
yxu
yuE
byu
yxu
xuE
ννν
ννν
2. ( ) ( )
( )( )
yAExAExAExyxx 222 1
2,212,
12
νσν
ννσ
+−=+
+=
+=
3. (a) Yes, (b) Yes, (c) yx uu , non-zero along the base ( xxε is non-zero which in itself is inconsistent with 0=xu along base)
3.2
3. BA 5= , ( )( )[ ]32 103021 xxyEB
VV
−−=Δ ν
4. [ ] [ ] [ ]234532
4532
45 244,113,1715 xyxyyxyyx ExyEyyExx −−=+=−= εεε
[ ] [ ] ACxyyxxuBCyxyyxu EyEx ++++−=++−= 44
1122234
413
4533
45 ,175
6. ( ) ( )
⎥⎦
⎤⎢⎣
⎡−+−−=
⎥⎦
⎤⎢⎣
⎡−+++−=
2
3
2
2
2
3
2
2
2
2
2
3
2
2
2334
316234
bLx
bL
bx
bxy
EbFu
ybLy
by
byx
EbFu
y
x
ν
νν
8. xgyxggy xyyyxx βρσ
ββρσρσ 22 tan
,tan
2tan
, −=⎟⎟⎠
⎞⎜⎜⎝
⎛+−=−=
