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Question 1:
Solution
2 2~
~~
2 sin (2 ) cos
F
y xy j xk
x
∂
= − +
∂
2
4 2~
2 ~~
4 cos(2 ) sin
F
y xy j xk
x
∂
= − −
∂
~
2 ...
Q2:
Solution
cosx R t=
sin sin
dx
R t dx R t dt
dt
= − ⇒ = −
siny R t=
cos cos
dy
R t dy R tdt
dt
= ⇒ =
9
2
t
z
π
=
9 9
2 ...
Question 3:
Solution
~
2
~~~
kzjixyF ++=
zy
kzjixyk
z
j
y
i
x
Fdiv
2
~
2
~~~~~~
+=






++





∂
∂
+
∂
∂
+...
( ) ( )
( )
2
2
0
1
2
1
22
1
22
9 cos2 cos0 9 2 0
18
9
9 2 1
27
z dz
z dz
z
π
π π
π
π
π
π
−
−
−
= − − + −  
=
 =  ...
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2b. sem 2_20112012_(ans_test_1_eqt_203)[1]

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2b. sem 2_20112012_(ans_test_1_eqt_203)[1]

  1. 1. Question 1: Solution 2 2~ ~~ 2 sin (2 ) cos F y xy j xk x ∂ = − + ∂ 2 4 2~ 2 ~~ 4 cos(2 ) sin F y xy j xk x ∂ = − − ∂ ~ 2 2 3 cos(2 ) (sin ) i j k curl F x y z y xy x y ∂ ∂ ∂ = ∂ ∂ ∂ − % %% ( ) ( ) ( ) ( ) [ ] [ ] 2 2 ~ ~ 2 2 ~ 2 2 (sin ) cos(2 ) (sin ) 3 cos(2 ) 3 1 0 cos 0 2 sin (2 ) 6 x y xy i x y y j y z x z xy y k x y i x j y xy y k  ∂ ∂ ∂ ∂  = − − − − −   ∂ ∂ ∂ ∂    ∂ ∂ + − ∂ ∂   = − − − − + − − % %% ( )2 2 cos 2 sin(2 ) 6i x j y xy y k=− − − − % %% 4 2 ~ ~~ ( ) sin 4 cos(2 )curl F x j y xy k x ∂ = − ∂ 2 ~ 2 ~ 4 2 4 2 ~ ~~ ~ ( ) 4 cos(2 ) sin sin 4 cos(2 ) 0 F curl F x x y xy j x k x j y xy k ∂ ∂  •  ∂ ∂     = − − −       = g
  2. 2. Q2: Solution cosx R t= sin sin dx R t dx R t dt dt = − ⇒ = − siny R t= cos cos dy R t dy R tdt dt = ⇒ = 9 2 t z π = 9 9 2 2 dz dz dt dt π π = ⇒ = C zdx zdy ydz+ +∫ 9 9 9 ( )( sin ) ( )( cos ) ( sin )( ) 2 2 2C t t R t dt R t dt R t dt π π π = − + +∫ 2 0 9 sin cos sin 2 R t t dt t t dt t dt π π = − + +∫ ( ) ( ) ] 2 0 9 cos sin sin cos cos 2 R t t t t t t t π π = − + + − [ ] 2 0 9 cos sin sin 2 R t t t t t π π = + − ( ) ( ) ( ) ( ) 9 2 cos 2 2 sin 2 sin 2 0 0 sin 0 2 R π π π π π π  = + − − − +  ( ) 9 2 0 0 0 0 0 2 R π π = + − − − + ( ) 9 2 2 R π π = 9R= 2
  3. 3. Question 3: Solution ~ 2 ~~~ kzjixyF ++= zy kzjixyk z j y i x Fdiv 2 ~ 2 ~~~~~~ +=       ++      ∂ ∂ + ∂ ∂ + ∂ ∂ = ( ) ( ) ( ) ( ) ~ 2 2 3 2 1 0 0 32 2 3 2 1 0 0 2 sin 2 sin 2 sin 3 V V V div F dV y z dV z d d dz z d d dz z d dz π π ρ φ ρ ρ φ ρ φ ρ ρ φ ρ φ ρ φ − − = + = + = +   = +    ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ [ ] dzdz φφ π ∫ ∫ − += 2 1 2 0 9sin9 [ ] dzz π φφ 2 0 2 1 9cos9∫ − +−= 3
  4. 4. ( ) ( ) ( ) 2 2 0 1 2 1 22 1 22 9 cos2 cos0 9 2 0 18 9 9 2 1 27 z dz z dz z π π π π π π π − − − = − − + −   =  =    = − −   = ∫ ∫ 4

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