Vector Jan11

8/3/2019 Vector Jan11

http://slidepdf.com/reader/full/vector-jan11 1/9

UNIVERSTTITEKNOLOGIPETRONAS

COURSE

DATE

TIME

EAB1 123 I EBBI 123 _ VECTOR CALCULUS

08 MAY 2011 (SUNDAn

2.30 PM - 5.30 PM (3 hours)

INSTRUCTIONS TO CANDIDATES

1.

2.

3.

4.

5.

Answer ALL questions in the Answer Booklet.

Begin EACH answer on a new page.

lndicate clearly answers that are cancelled, if any.

Where applicable, show clearly steps taken in arriving at the solutions and

indicate ALL assumptions.

Do not open this Question Booklet until instructed.

Note : There are EIGHT (8) pages in this Question Booklet including

the cover page and Appendix.

Engineering Data & Formulae Booklet will be provided.

Ui:.ivers: ::- Tekno-icq: ?aTRC);;S

n.



a..

EBB1 123

Find the work done by a force F = 5i -2j *:i in moving an object

along the line from the origin to the point P {1, 3. 5i-

[3 marks]

b. Compute the magnitude

the bolt at the point P if

I and F is oo".

Find the parametric

planes x+2y+z=l and

of the torque exerted by the force F on

Fl:sn't, lFl=40 N and the angle between

[3 marks]

equations for the line in which the

x-y+22=-8 intersect.

c.

d.

I marks]

compute the distance between the line Lr through the points

P (1,0,-1) and Q

(-1,1,0) and the line L2 through the points

R (3, 1, -1) and S {4,5, -2).

[7 rnarks]



a..

EBB1123

Determine the velocity and position of an object at any time r. given

that its acceleration is c(r) =ti- +i j*cos2t*, its inibal velocity is

t(0) = i + t and its initial position is i(0) =;- .

[6 marksl

b. Calculate the arc length of the curve

V(t)=(2cosl)i+(2sinr)7- +f E for 0 st <L4

d.

[5 marks]

Find the principal unit normal vector (N) and unit binormat vector

(B) for the curve

V (t) : (sin r) i + (.,8 cos r) ;- + (sin r) f ,

[6 marks]

Find the limit, if it exists, or show that the limit does not exist.

rt+y'llm(r,y!+(l.-l) X + y

[3 marks]



a-. The temperature T in a region in space is given by

T(x,Y'z)=2x? -ry7 -

A particle is moving in the region and its position at time

x =Zl2 , r:3t ,

Compute the rate of change in particle's temperature

P (8,6,4).

EBB1 123

at the point

[5 marks]

[5 marks]

tts

b. The temperature at the point p(x,y,z) on the sphere

x'+y'+22=l

is given as T(x,y,z) = 400xyz2 .

Use the Lagrange Multiplier Method to locate the highest and

lowest temperatures on the sphere.

[6 marks]

c. Evaluate the following integral by reversing the order of integration

d.

3l

I I'r' dy dx.

o1-'

!l[4 marks]

Evaluate the triple integral by converting to spherical coordinates

r Jr*, r

J I I t"+y'+zz)dzdydx.r

-,-...:-=-

_, _Vt_r, Jr.+),.

(



EBB1 123

3. a. The temperature f in a region in space is given by

T(x,Y'z) =2x2 - ryz -

A particle is moving in the region and its positlon at time r is

x:212, /=3r, z=-t=-

Compute the rate of change in particle's temperature at the point

P (8,6, _4).

[5 marks]

b. The temperature at the point p(*,y,2) on the sphere

x'+y'+22=l

is given as T(x,y,z):400xy22.

Use the Lagrange Multiplier Method to locate the highest and

lowest temperatures on the sphere.

[6 marks]

[4 marksJ

d. Evaluate the triple integral by converting to spherical coordinates

r.[Jr

{ f !r'' +v2+z')dzdvdY--' -VI-r- r,i r'+)''

[5 marksJ

c. Fvaluate the following integral by reversing the order of integration

31I I nrt dy dx.J)o l-'

I-

V:



EBB1 123

4. a" Find a potentialfunction f for the vectorfield

F{t,y,z) = (;,sin z)i + (-rsin =)i - {-r} cos:}i

suchthat V/=P"

[5 marks]

b. Compute the work done by the force field

F$,y,2)=eY-i +(xzeY +zcosy)j +(^yrjt +siny)F

over the line from the point P (1, 0, 1) to Q (, t, O').

[6 marks]

c. Construct the analytical function U(x,y)+ iV(x,y) for the function

U (x, y)= sin,rcosh y + 3x" y - 1'3 .

[5 marks]

d. Show that the function

U(x,y) = yz - x2 - sinh 2x sin2y

is harmonic.'

[4 marks]

5



I

EBB1 123

5- a- Use Green's theorem to evaluate

{tr'l dr+(x2}d1'C

where C is the circle t'+yt :4.

[6 marks]

b. Use Stokes'theorem to evaluate

{F'*C

if F(x,y,z)=r'i +2x j +rt E and C is the ellipse 4xz +y2 =4 in

the ry-plane, counter clockwise when viewed from above.

[7 marks]

c. Use the Divergence theorem to find the flux of the vector field

F(r,Y,z) = ZxY i + 2Yz i + 2xz E

with outward orientation through the surface of the cube cut from

the first octant by the planes x =1, y: I and z =1.

[7 marks]

-END OF PAPER-

6

APPENDIX



APPENDIX

1. Vector Equation of a Line

,=a+ti

2. Scalar Equation of a Plane

a(x-xr)+ b(y - yo)+c(z -zo ) : 0

3. Equation of the Tangent Plane

4 ("-"0 S+ Fr(l -yo)+ F,(t - zo) = 0

4. Total Differential

dr=dt dt*o'd,0x Ay

5. lncrement in z

6s= f (x+Lx, y+Ay)- -f (t,y)

6. Torque

.=lrll,r'lsine

7. Green's Theorem

I p.a, = ff (gg -9!\*c ;rn[dx Ay)

8. Curvature

I 'L:l ,. '"(/) I I r,(r)irc(r)=l

'-' I or rc(t)=1,' t",

l--'' lt'(r)lI''Q)|9. Cauchy-Riemann Equations

0u 0v 0u 0v

^ = ^ , .. -- - where -f (r)=u(*,y')+ iv(x,y)0x 0y 0y 0x

10. Laplace Equations

A2 u O2u 62v Azv-----:-*-=(l : -0Oxz Ay'v ' Ax2' Ayz

11" lntegration in Complex Plane

I f uv,=(L "0,-"n r)* (f "a'*,a t)i

12. Cauchy's lntegral Formula

EBB1 123



EBB1 123

-?

.f'1^

=P

{

f

CudF',

ni f (.zo)

f(:\ ,

t to

i +Qj + RK

VxF =

jkaa

Ax 0y Az

P A ft

Divergence F, F = Pi +Qi +Rk

Div F= oP *99*9!0x 0y dz

Divergence Theorem

JI F.dF= ![[ at"F av

15.

16.

17.

18.

24.

Stoke's Theorem

I F. a, = II cartF.nds

Arc Length

b

L = Ilr'e)ldt

Surface Area

19. lntegral: (a) I"&)'.[#)''du = eu +C

r(s)= Il dA

(b) I.f';'d' = %.l i *"

+ "'/tn1' * ^fF * f 1 *,

TNB Frame

f (i =fq, J(r) = !9 ,' \"i''(r)i

' " \'rlf'trl

'

8

Eg1= ro) * lr(r)

Vector Jan11

Documents

Transcript of Vector Jan11