Issued on: WEDNESDAY 22nd APRIL 2015
Engineering Tripos Part lA
Solutions foF }lathematrcs pEper
L (i) f(x,g) = x2us
,*
,)f/ix
vt.
vfr - 4rag
= Zxgs I; vJ:i,r,)xl = D/Jx t,)f/,)xl = igII
,)f/nu = s#Ua; a?f /aq? = iiiat_l iarl;ui ri*=u\.:'
=
,)?f ,IJgax
= i/ar1 tar/ixl = l0x
.t?f i axag = ,],r,)x taf /agl = loxg{
.;lr"lH
10
i,i)
f{x,g} = x sin U
of,rdx = sin U; i\iAx2 = dldx ldfldx] = 0
,)f,/Dg = x rlos'J, ,]:f,/AU2 .)/ig tJf/Agl -xsin
=
=
,izf,t"lgAx = ,i/dg [df./,]xl = tos I
,i2f,.',tx,)g = l/dx taf/.3ttrl ros g
=
7
5{-O8
-}_-
U
-39
?
l==
I
30.oo
,}
.lt
a
b
t26-08
Approximote vslues are as follows:
,)f/u1:+ st the centre point = i [(zr,
=
,)f
i Jg
"i2;,r.1s2
lza zu l,/?ax
at the centre point = i [(zs - zr) /ogl *
= lzz z* Ii2ag
at the centre point
=
i [(2, - z=]/axl [zo
.f f ,/,)gi
- z=]/Axl * [(zs - zo)/txl )/2
at the centre point
o:t-,',]gox at the centre point
=
=
*
(l)
I(k zr) logl l/2
(2)
[(z=
- zo]raxl ]/,rx
:, - 2zrlr'(ax)z
(-1)
- a=,t/agl - [{2, - z*)/agl }/ag
lz* * :- - ezrlr(ag)2
[ [(z=
{ [(z= -
r,ir'laxl - [(2, - zr]i2axl
- [=. * zr - rr =rll4axag
(4)
]r'2rlr1
r'S1
,
Issued on: WEDNESDAY 22nd APRIL 2015
,.iif,rrtxdq at the rentre point
=
i[i;.: -
zr]/Zaul - t{2, - zrillrrgJ }i2ex
- [=, * ?7 2., - zrl/4axau
{6)
Substitutirrg vaiues f or z:
,:t the centre point = {ll6.0g +l45.ll - 260i/0.1? lI0
=
,]:f i ,}g: at the centre point ( 126.08 + 134.08 - 260)i 0.12
=
=
;ilfi,}ul}x ut the centre point : ( rqg.3g+,il2.3?- lt9.e?- r40.e9)/a{0.1x0.2)
,i:fiix:
i
= li)
tJe
-lo. aT
f.o* [5) *r^d- (e ) thq.t these q.pprorl^o{,,0^s
',J,i*.A
2hl' pa.(,'o(r'^'J
#" are a(way s o?unl. (fA.
.(;k ,*S.(r"s q-tre Ll,u.o-L provl ,{.inq th" Li'.^i{s q.s At+ O ar,A Ag+
aFe tke Sa-u,te , 'no ,r^a-tte r if, ,^rhi.[. orde, t(..t aFe t,"ken.
T[is ca^ be s[o*^ to hoLd if the 1n4,,'it<J pl"{,h(, o.re cort{l,rr,ors.)
Eee
q.,ad.
O
[The dato was constructed from z = lOxJ
i.
Fr-rrwhat value of n is B = tn exp
* xu2l.
i-f/4tl a solution of r-he equation
;i/ar tFaB/Brl
=
F aalatz
LH5
= tr, exp {-dl.4ti [-irl+tl
r?a8iar: -Q.51n-l Fexp {-rzl.lt}
;B/,]r
,i/i:r IFii8rdrl
:
= -r] 5tn-r
r)
!g1n-r
-
exp t-Fi.ltt t-tr/4tl
l.Stn-rfe;rp
esp t-Fl.ltl
t stn-r exp
F
d
-
f
i-ri
i-Ft+tl il)
PH5
,iB/;it = tn exp t-Fi4tttF,/4t:1 + p1n-t exp {-f/4t}
r: il8/dt = 0 25tn-2 r+ exp {-r:l+tt + pln-l F exp {-Fl.ltt
tlonrparing
(l)
t?)
unrl (?), n = -1.5
{.
taj w
li{i
=
= icos u sin,}) isin B sin,}i(cos 6) =
::r-i ij",,Ti'ir8l*
= rr0s tB rin: ,} r::Lrs ,0
it,i
05 sin
,i,+i'i'jEl. = i-Jii-::in B gin 0) * :iiico,j B sin r}i * :<Ui0i
-iinz 0 r_:ss,l,* ,:oszg sin:0 c0s g
= -!r# B
=
28 srni,tr cns0
uL15
iE :;in2,} Cui,l
4ti
Issued on: WEDNESDAY 22nd APRIL 2015
Nu. axf
5
re
ss
r,r of
io
the
-foa rn
r Q(rrv)c[y
P(r,,,J)d"
is c\ per{e.t otifi€,.<^tio[
Sod^€
f,^^.t.'u^
tJ P= H
(r)
a,,.rt
a=S
-f(r, g) so tho-t
At= UA,.H*, =
P,,Lt
fo,
+QJJ,
tt^e i^k1.^[ ato'^'1 o\ patt" i^ the x1 p{otne
i. ssa,3 ot^d tu{ €,'.lo^1
d.e pe,,,.J-s ot,[1 h tke e r^d po i^ti . A
^.
.oJ.*itio^ '+-" (r ) to 6e *'funfe.t A^l"len^t-.'o( is tt,qt = H ,
wLicl^ M€(t^s tLo.t
o
^
#
Qire.vt dtn= TJ.s + rdP, ,tote f,'"st thot d.s or,A J.p
*To sur.^.at( ?i""{ I tias o rt i-he nil Lf L.,u^d s ile
etre
70so T o"r.A v ar€ i,.ap(ieA {--tnc{,'oitS .1 s a.ncl P,
tdt us * "i'(e
d.[^.= T(s,p)[s + v(s,f )dp
T(e,,.\^,e
Lnd ,^ite
F7n"('"3t[.
have/bh\=T
a^c[f*)
I *'*'
(;P ls =v
A pun(,)i;',1.,
'DT\
/s
Dp
t-
|
\
-
d\
Dpbs
Ds
F.o^ oq= h-Ts we hou<
)p
dJ=
'
dh-T/s-s.,tT - rdp-tJf.
we o bt-ol^ I
af oi<
As
rlr
\>s )p
*Ll
b'h
)v rr [ ->t /r
t,
r
< l.-
)t
\>r ),.
)Vr[,,r)= L-3i
6.
Rr.
cr.
(()
unit
L=
)
rre
utor
(^Ei
b- the directio^ol
{r ))
),e iur.-[iv" ]sT = L ,V T
lz
t-? . g.,*). = )lT = [,{t - z)
lt
^le
Issued on: WEDNESDAY 22nd APRIL 2015
t-
/b\
rt
t 3f)
L-/ tb = {t
\{e-,,"1.
)
Lc)
tt.ol'.
-7,
J
raJ,.
/tr,d
DuT
=
-
tf
-tl/r'tlt
=
-t
[^lest *[leut is wf^ein b = vr
3
-l
DIT = vrl= ,ti*'
V*=
\
dqr
f tvrt
= (L-3i)1,fi.'
uYh
i t bsi+
by
bz
+ 4xz h
= C;-3b-f)i
Vw ir noavrnaL to cr t.v.L s.^rface w (x, \,2) = cora-st,
'ThnE
o.t te, T ,z) ,
tb@ ywa h X iv-" b,J
b^
,,
.,
vw '(r r") =C
ircctor i,n
tuhere !: ).:oJ JI z\ ct^A Io is
"^,1
.)__r
the
t-t-
-$
ta,r"-jea
1r,-t
t
,r)
I
tatute
Vw [t, -lrz)
a
3iv"s
lx_3b r
= 1t -31- + th o.^J
a/
8z
g
zrr'I
.ai
\-----
I
./
l,
?
o
t
: rt1
)r=
i
-r 11
\\
\.
\I
I
I
The thich btr."&. Lines
t[,,e z=O cowlout15 coffe5pov,J,'nq
Bl consiot,eri*j tr'.. J
to r=O, I= O "lrt 3 ^ l-?112,
,(
t:"J^ of e".1.,f".S. .i,r z=)tg(z-r,-2-g) i^ eorL re3io6
o..re
lYn"s rlppro*ir*o.te s f.apt of
Cout^[
o
ot{rS
+_
Issued on: WEDNESDAY 22nd APRIL 2015
I:
- :t - lgi
= 2l;U - :i:,J
iri f f erenr. iat lng parr-i al lg:
iiz,rilH = irgt I - :c - l_l)
:irJr.I
- Ii;g:
i tt
t'_.'l
ilZliiU=fitl?-x--lg)
t1t
5tation;lru points of,cur when:
gii - H - !J) = 0
i4i
i;(2-:i-4gi=0
l-rl
Srrluing these simultanenus erluations
Fr'-rrfi i4), U = 0 0IH + U = I
Frr:rrr
i5), x-=00f x+ 49=2
There ErB f 0ur Et-sti0n6rU point-t: i0,0); i,2,0); i0, I ); 12/3 ,l /3)
Istthnee ctne Sad.dles aruo( tha
Fro* cotttours w( see
rt..ct).inauuta. AL{e,/^a.lin" [r, proceectlnj ctr,.a[yfi.aL{.y i
4fi
tkqt
it ,.
,
&=-2q
-4x,
17\, b'3 = L-Zx-4u
--), &=
J'
Dt"
5ro5
;;Ldrz >12
).fi,.e A = Drtr
dl,
Frou,^. P.5
Point
(0,0)
r.:,0)
/t)
[]a]y/
\z
'
of ^otLs ,\oto. book w(
..')
t(
0'zl dy.l a2z/dtl2 dzzltxtg
.1
,(eJace:
A
De*tctioo
0
D
2
-4
S.rddle Point
D
-B
-2
-4
Sadle Poht
0
-.)L
-4
$addle Pofrrt
-213
+4/3 Mexirnf,n
(o.t )
i2/3.r
r )rz.
-sls
-213
a
[,
The f igure shows a constont v Iine on the surface:
Constant v
line on the
sUrfECe
r(u,vi
r(u + au),v)
/t
i.i
o ,tr'
.J
Issued on: WEDNESDAY 22nd APRIL 2015
Then the vectr:r PQ r(u * .{u,v} - r(u,u}
=
rhe'rectorPQ/,tu = [r(u + ou,ui - r:(u,u)]/au is in the same direction as pu
in the limir- Ee au +0, the r,,ectcr pQ/au becomes a (non unit lengthi
i-anrlent tr: the L:Ur!'e at ErOint P.,rvhereaS the RHS
becomes crr/iu.
A similar argument for a constant u line shows thot drlDv is a tangent
to
the cnnstant u Iine on the surf ace. Thus ,lrl,)u and drlav are tangen-ts to
the surface, but nr:te r:arefullrl thot theg are not unit tangents.
For r
=
(u2
*,",ji
+
2uuj
+ {u + r72ig
itr,rrlu=2ui*2*Jj+ lk
,]r/']v= li+?uj+2uk
itt
(!'l
5ince tliese vect0rs are tangents to the surface at the point r{u,v),
their
crtrss product is tn the r1irection of the normal to the surface at this p,int.
Then i)r/,)u:<clr,/elu ={.4v2 -2u)i +(| -4uvj
(3)
Fur u =
nnd v: -1, ir,i,)u x crr/,;rv
+
= 6i -3j 6k- The length of this
rir:rll.or is g, go the unit nurmal is H (Z,/j)i
-(l/3)j , i:/gjf=
+(4u2-2v)k
-l
The corr:-:tilrrt u antl uonstont v Iines on the suriace interseut orthogonullg
when the tangent vecturs tr: these curvgs are orthogonal. Forthis to
tr,:pBen the dot prorluct r:f the l.nnrlents to these two curves must be zero.
Frr:rn ( li and {2):
(4)
= 2u+ 4uv + 2v
Fur u -- v: -1,,)r/du . ir/dv = 0. so the curves intersect orthogonallg at
ihis particulor point. Eut not at euery intersectionl
'iritu.iritv
14cY
(,