Tue Slesu'e pierd yDar pe qiven
e(«iy) =324Y t22y . oTt y e
Pind e ele crie
point (2, 3)
-4 42c
Eetrie tietd e0 at peur 2,3).
fluy,z)
+ Ou dz
y
Tötou deiVoi ve
(a) TotaL derivative
du/d +
de de y d
Let 4(2,)-C du
da
ay
gctr,tz)
f2,y
du
du at,
u.dy
u:224y 2
dt
(2x Y2at) +(2y(2a)
4ay
4alzt+y)
4al a2.t +2ot)
Y= 20t +2 Y
du -2
ày At
(22 2at)+ (ay2a)
4at 4ay
4aca?4 b?) t +(2ar+2]
ab.
(2%.2br)+ (2y. Qb)
2att24)
a24br2) + 4b(
ibrt
dy
a.ry3 3 Ainy
3x4'2 Ainy
u=.2-y+siayz; y-e du
dsc
du Du(hedu.ay
az
duy 2x2y+ zcoAyge +(ycoyz)L
du x -2ye+zconyie4 Ycoyz
2x-2e
u: (2x-3y, 3y-tz, ma)
3 ôy
oy
3Y-42=s
4z2=t
u-fr,st) ,t are
du (duy
dy d
bu . (-3) + ây3) +
dt
du . ds
dr ot dz
as
(-4)+ udr (4.
u ( 4 , y-z,2-)
frve u + du =0
Y-Z=
-t
fiT, s, t) , .s, ttare yy,z'
uz
du, du
te dz dr +du.ds
de dx
dr dx
flr, s)
u Ou. r + u. @s
ôy
youy
8z ôz
()
X
(*)
u
h-utlz
w
h-w(-wzhut
b-ulT-u)
2wt
^ty)ut. h-wlw)
(;2t,
+
2
(y². zx) au +(z?-y2)8u
dy (22-y) Ou
u-f(r, s)
S= y 24+2z
Ou .ar
Or
Qu (2) + ou (2z)
dy
Ou (22) + ou (2y)
(y2zx)ôu t (x2yz) au +
a(a2))+(1?-ye)l au (2z)+au (21)
-(y2z)âu (2x)+ Ou (22)
2zzUrt y'zU
axy2Ur -
+zdu ?
Oy
du Ou.dr ou.d Ou. ot
x2
du - Ur- lz)
X2.
Qu.ôt
-Ur Us +O
y2
y2
u. r + du.ds
ds
- Uy (0) + Uty) + Ve
xdu yôu +ZQu +
-
yu
) e-24 ((a£)(o))
( ' + (af
Y e ny
dy u
f f e
fencOf +f)
sinu) + af (4e cos)
/ Of.-e4
2
sirtcesF
2
cos2)
-eu Sin2)
2&íngces)
(-e?
sirsce
+ 2 Ôf f 205.f Ainocee)
Sin21 +
Pind We & ode at uahichte aiea
a itangie
giveu instait oheu
the heetangle ae 4ft aud ad aei inereasiney
he rate
et the
sides oy etangle
b=3
Aerea
AS A
2
bl-s) +4 alor)
4-f4 2 G.
. e cot C % mateviala wsed n eontuetion frejet
is depeuden o the quantity mateial à o.
the phie per uit Pay CP. a) - Pa witha| Operwni}
requises 00 uitd e maerial an
e pogect
pries iicreases by 22 ber hit nohile th¿ qu auty
the oi Ho doea th e total cert
needed dereases hy
thange
dC
-|0
2
dp
I0(-10)
dC
+
2! 3!
+
f'ea) f"ca
z2f"(")
21
21
(a6) (y -b) fy (a,b)
Jtay fta, b) +(7-a) (as)
t
Cy-)?fyca,b)
3(a-a) 2Cy-b)fu, a,b)
fe-af (a,b) + (4-b)?fu,(a, b) +(4-b)ry
3| + 3(x-a)
pouoen ag(z-a) (y-b)
MacLares
(aso, b 0)
neighbeuheed y the pt. (a,b)
he
abo-e
UPTO Tnd Doqree teir
deqree
fa,b)
x(a,b)
fy(a,b)
2yl4,6)
az(9,5)
fy Ca, b)
jyy
frmuta- tyy Jy
-sinz.f.os:y frzy)
cosy,o(x
flay) -siny :
1+
fay) coAX
- sinN
siny
cosx
xy+34Q -1(24y?) 2
7(b)+ fa,) Lesy
(-x2 tale,0) +
fyy(ab,)o 4 fx
JyCaI,tb3)e:-6-2
Cafl, a,b) -10 ylo) lo,)0-0{tfo,0) =
Jylo,0)- D
b) -2 -y?) (z-a)fu
2y - +
t
fny 4 n 274 Ca,
fy b)
Ca) +(y-b)fy
lo,
bo)
ae0,
o)
+y, (a, b)
lo,o) o,o
7°y+34 -2 + +442))4)+l (°t-a)+ 24(2l +y2co)
2|
2y +3y-2 -42 +4 4 44+ *-2°+2y t
a ;b)
fca,b) 1
pla, b) | |
fy (a,b):D
fay
fyy0
(a,b) +4)fla,b)
f,
exy): fa,b)+(z-4) 22yfayCa, b) +y2fry Ca, b))
2xyli)+y)
+(94) ( 0) +1 /x?(o) +
(4)()
A(z,y) 1 +
ftz,v)
at h e
find l .|,0.4)
-4
224y2
y 24(1)- z(27to)
x24y2'22
(x?4y)2
fzxli, 1) 2 12
(z49)? (74y)?
fyy(i,1:-2
Syy -24 2
(12+y)? (242)
(aa)f ca, b),4 y.bfy(a, b)
fomala: fta,y): fla, b) + (a,b)+2y fayla, b) +ýf,)
(°fa
oe0y)+6-0(4)* 2
4
aylo)
* - y2
)
4
, o.9) - }+0.9 +(1-(0-9)-+
2
0-2
ten obt. (o,0)
Q- txpaud e' iagct4) wpto Mrd äegree
e'glt)= flo,o)
2|
Lr
3
(00)122Yfry
43Y (o.0)
31
+3y'x fv lo.o) +y
f ) -eog(ea). floo)
f,lo,0)- 1
1+2 tbmwda
fco,0) (0) +
0t Z ) t y
P(2)+3y4) 3y
e -) talo,0):
(14z)
Jyyloo) 0 0+x++-7
Loglux)
(213.$-3x24 +5y'x)
2
fzy =-ey fzy?
(-+)2 +22
fyyx1 y-a?. 3
fyy2= e
+2e taul0,0)=2 - yt y 2 .
-eY(-2)
(Hz)3 (I+z)3
Expaud e 64y abt. (,m/4)
fi,/4) : e
+allg tt/4) e
fyesiny
tzy -e siny
f,1,T1/4)
fy -eesy fvy(,7I/4)
22 2l +Xy-h) y
e"coay e(7-1)e +(Y-T/4)e*
2(2)(4-TU4)e -je
2
yt abt (l, 2)
Expaud , +34
6 12
fe3y2-42-9y +26 fu,2)- 1+12- q-|8+2
29
fy el,2)+3
fy0 fxxel2)>2
+)()+2(Xy-2) o)
(3)
t24 (x-)t-) + (y-2)
I2-74+3Y-otz)?
|13-7x43y + (4)" 3 4-2)* A
ft)
Naya formula t hxt 2hk fuyt kft
f(x+h, yaY):f(ay)+hf tkfy 21
(x+h) (Y+k)
. expaal (xthy+k)
fu (+)
x2
fy-o fy c(+4)- y
(r+)2 )
fay
JacpoBim
T(u,v) = 3(u,v)
T(u,V,w)
a(x,4) atu,)
(2) If U, ane
(u,v)y d(,s)
(xy)
Hhen be
ether wiAe u,v ane depndedent.
a(x,4) vaiales
(4 Jaobian tt
)0
( u,, 2,
( F,f2)
a(u, u2)
(i) ter u,,w me
6(u,,w, x,y,z) lu,9,w) )
Vg9 , y,z)>o -lue,
8(*,4,z)
Su u du
lu,u, w)
(,Y,z)
ou
alu,o,u0)
ax,Y,Z)
CsCz-G
z(z-y) Y(x)
Yt2
| z(Z-y)(*-z) -}
()(-z)(z-y)
-(u,gw)
2
(*)Cr2)(z-)
d,Y,z)
h-(n'nn)
^'Z)e (2
-
-Z.
(
hz
e2tc^ten eztehtex
(x-2)(z-h(h-x)
(onn'n)
x)e (2'h
(3y'y3)9
onane (2'h'%)9
(2h')e
(z'hbee (2'Ka)
(onnn
(2'h'x) 27
'nne
-onga ) (ron
zo1-omya) a - 1
nmtonn-An-z (ngnnzn+
t(n-n (an)em (n-a)n+
(2'h')e (5'e
(n'n'n)
n-th+x
(n'g'n)e
(2'A')
(z'k'x)e (-)
(n'an)e
(2'h'z)
(on'n'n)e
+() r)
sinu = sin l sintx iny)
sinu Ces (siny)+ cos(sinc) y
siu
Anu =
sinu
n (
(h'x)e
(2'hsue (3/heje n+h
h (a'9e tht
(o'4)e
gush? t,z
(e'a)e
(hx) har
zx-zh-+ )e)
(22'x
7
+
277
N
1e-eint
à(,v)
lu,)
-
e"sinw e e
-) 8(x.y,z) X
alu,w)
(u,v,w) à(,y,2)
a(x y, z) tans wsecy o
a(u,v,w)
( , , 2)
= 7'y?4z?
24 2Z
a(2,y, z)
3C3-44
2ly-2) 2(2-x)
22
2(2)(z-)- a (2-2) (y-)
(w?=t2u
e ubic ep.
are-he t
(A-x) 3
A-y)34(A-z)3.2
3_0
A -he
In
(u,t)
T (u,v, w) (x,yz)
3 -32 )
-3Ay2)
+ (a3y3.31y 3z22) -0
(A3-z3 - 3az -22(2+y+Z) -3AC
y3423) -
3-( A(y24z2) eo
2(+4+2)-
A-(34y3473)
3
U (wtwlty4z2
(y'z?)
Fa wwsu-zRy2-22
wsw -d F3- usw- (z4sz)
3
