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

, −=⎟⎟⎠

⎞⎜⎜⎝

⎛+−=−=