MU LTIVARIABLE CALC ULUS

2&MATH-2063

FINAL EXAM

Na,me:

Student Number:

Answer all problems.

All problems are worth 12.5 poiats.

SHOW ALL YOUR WORK. ADD PAGES IF NECESSARY.

(9 pages including this cover)

,

3.

6

7

Total.

Begin +

1. (a) Find and sketch the gradient vector field of the functionDetermine the strea.mrines of Vr i, the form f (*,g): ""{l;,

y) : +(n2 * g\.

// t"t/ / {!

".fll( il{ t}\ t r.\t\\ \,rt

v -.-J-

r\,\\'L$ L \,'512 \ \l

\Vl = (r,-d) \StrCa.rrralt a,.a :

4' -- * 4-r -- - *{ '>a&v- ,, - {x; nrV 1=ba-V a)brclR

', 7 )lt1 ?tr,,r

ra!

t' F3,Hili",{: It-T,1,[

F'a,'where F : (u,-,,u)and c is given byl'+

^JI r-. dt =Uc

n r/anu t =

3

2- (a) Check that .F : (l + *y)dui + r2e*sj is consenrative. Use this to evaluate theline integral 1"F.af, where C is the curve r-: (cost,2sint), * e l0,rl2l.

JueY{+ ufi"'1- x e"I

(,"t()I"X = o

T\(", {-(*,!)= xexl+ {''f^)

Bt tu {r+ x{)c'i r }}" (.'()'*( :D fi=

(b) Find the centroid of the quarter circulgregron of radius a.

\"-- I \-l'o"^t cL&VTi cr JD

e.tv.LL t^.,c -l 4- i = o" ( tn t

'

**,1 f-= T=Yr/'''

' 2 e'A" [.s"-,}- \ fL -ta

$fll ,I

=S

n fYu

Jo

4TI

Carr-t

L

3. (a) Ewaluate the line integral $crEda+rzyzdy directly, where C is the trianglewith vertices at (0,0), (1,0) and (1,2). Then use Green's theorem to evaluatethe integral via a surface integral.

^= ^J ,'{urt1f

l-tdrt = Ju

o

I

?oI

Jr

?

f ,,'+.x,(8"\3]'a'

,, l, + ,iTr, I

G*-" t!^Lorct'"'" -5 -\

jF,ai , [j v*i'i_'A = iJ ("t'- ^)

.Jr ji ac' s

J'il't-'*ul^'t\1"*'=.|5(b) Use Green's theorem to evaluate the line integral of .F : \l - y3, p31 sv2) over

the curve that is the boundary of the regron enclosed by the parabolas U : 12

t ui , --,)

a_'' \ drJlats/{ )

'\ &. I e'./ qd

t{.L= +l+

14 lD

X=! = X\ :5 X:

$ i.*r = Ji e??\3 t' I t{r(r^*r9) J IJ \o xt

andr:y2

tr3

4. (a) Is there a vector field i on IR3 such thatcurl G: (rsiny, cosyrz - W) ?Justify your ansvuer.

B-ccor-n S'V^Q =o *.

V. ( xat't , t,a, z- ^X > ' 5''^ f, -6'^ { +r

-(1^-'*-y.^ otu.u- ; l.l-1 *c^for $'U e' s.^d^

V, a a (xs,^Xl cu\tq-- xtr>

(b) Verify the following identities (i: (*,g,2)):i. curl r-: O,

. ii. f hr :i/riiii. f(1/r) -- -r'fy3.

= | *p

,hr,*

(,1 d=L'r{,") i $xi = \;;

(,,) i0,^t = ()r.,4[ rDz') L'"

( tr,; V ( i) = 1 or, 2d, ?- ) {},t;;: t ;i1,. (*'Y'=)

( >,'1't z" )

lA,L\u'{ Ot' \=

(oro'o)

{-?-l

nl t', '1" * '-"*1.* r-) -- ( t, !,

;->.

( x-+-q'+z')

F/ na.

6

5. (a) Find the area of the part of the surface z : 4 - 2r2 + E that lies above thetriangle with vericies at (0,0), (0,1) and (1, 1). What is the equation of thetangent plabe at the point (0,l,gl?

Ats'r = j[ dD

^lzJJ'/

I + lbx' * I d x .I.r

d, Jl

il; (l-r * &x-I+

dy\,

k>"

\a,,- f '- qJe \v0

=1,'*l\l I*n.A- 1ta',,'

6. (a) trvaluate the zurface integral:{[ru2 12ds,where ,S is the part of the paraboloid a : 92 * z2 given by 1 < r

7. (a) Use Stokes, theorem to evaluate {"F - ar,where F :is the curve of.i"tu.."*io.,*ffi"Jhyp"rbojo id z : ,, glr:trr,ry) and c*2 + a2: 1, oriented 6"i.;;."il.,.i.o,^, n^- ^r - - o2 *ia tn" "yri"a*

* ::r: ,, "n""i"a;'"#;:X *H:f*:f :r:;

th G*".* *r^