MULTIVARIATE CALCULUS
, A"- e><p/;e_;_f- 1~-d-i'on + fwo
---~ - - -____,,.,,_- -
V~
-= --
K 1
'f 0vt<>- ;"ct,, em~+ v~ Ow\_!
(A_ Ap0vc_~+ ~ ~J _ z = fC"K ✓~
Ovv\ , e_,-pk,.,,J ~i 'o 'fc + ()(_,'/
L iJ + -f:ili'ovt, : CA_
A f"""'-&'""'- ,/,) sJ +, f0v-cl +, /Jc_
~ J_, od- roln+ (_XI '1) ~ t CJ--
0
°'- rci" 't l°'- ),) ) it "To <2A/0 ou,J,;~G
..s JL -+ Ve... cy\_W W\.,~ Q.rL. E.. ~ e,o~ f OY\_ ~
°'- rn_o_ b s~ -- U
\ f ('><-J'-1) - LI z E- w~ e_ v~ o ( {x_- 0--/ <~
0 <1-y-b/<S
C rt.- 0 < '/ _-6j-y
QS--___°'j_v _+_( _
j'_ z 2)
~ 1 (x/1) = L
~ J~) ➔ (°') ~)
~ ~ ;~ -~ lo ~ ~ DtL-
-5~~ ~ ~ ~kw-, 0<-/ '1) ~ L°'--~.
'P=reJie.i ~ ~ :
,0 ~ f(-x/ '1)± ~~ d( ,.__/ y) = J__ + 'YY L,
(){_ ,"i) ~ (G\ ,, b) (_-x., "1) -) (CA 0 I
0 \"'>=
~ -+ c~ /1) x ~➔ 3(_°'c'}6) 1- y_ Yvt ,
( 'X / '1) ~ l b)
(A ,I0( /1) j
_ k ~ f ('>t 1'i) = k~
0 J~k_f('»v
(2<- ,, '/) ➔ (o.};)
lX /i) ➔ (°' , ~
-f ('><- ,'i) J_
@ ~ fO
~ 1'f) ➔ (°'- ,b) B(~ / 1) - 'M - 1 l'V L
i°~ wi+lc, 'Yl.0
® -Jr re °""-J_ S ()Jr Q__
~ ~
~ ~ "'-- f'-darL ~ J s f d
p=vi U l% ~
j,: ,_ _ +C><:1) (Is = J_
%
' n 1
(c(x/'I_) ~ (<>-,, b) ) o.... ru..o.X___ ~ he re
- j O) + I <;, - -S-
~ L (?( -- 0 9(_
(-x,i)~(o /o) fi -J 'i
= !:0 ➔ (_o ,o)
L x{r,,- +rr)
= 0
, Jf-
= 0
LE½ 71Q)+·
~ S ho·w -tiJ ~ f( ~ /~ e_~ ~ -~ .
l!'-/'1) 7 (op)
1.,.,--
cSo /'D'l,:-
::; Let y = "YYL-, x_
1/m X 'j ~'VYl
-
-f (x_ /'1) = V
I+ rYL'l,-
·kry'
'V
+Y YYL'L- -
~ e_Q___. i_,m,, -f{~ ;'j) cLQ/) YlO+- ~xl/J+.
(_'l{ ;',) ➔ (9 /o)
Let /
, A"- e><p/;e_;_f- 1~-d-i'on + fwo
---~ - - -____,,.,,_- -
V~
-= --
K 1
'f 0vt<>- ;"ct,, em~+ v~ Ow\_!
(A_ Ap0vc_~+ ~ ~J _ z = fC"K ✓~
Ovv\ , e_,-pk,.,,J ~i 'o 'fc + ()(_,'/
L iJ + -f:ili'ovt, : CA_
A f"""'-&'""'- ,/,) sJ +, f0v-cl +, /Jc_
~ J_, od- roln+ (_XI '1) ~ t CJ--
0
°'- rci" 't l°'- ),) ) it "To <2A/0 ou,J,;~G
..s JL -+ Ve... cy\_W W\.,~ Q.rL. E.. ~ e,o~ f OY\_ ~
°'- rn_o_ b s~ -- U
\ f ('><-J'-1) - LI z E- w~ e_ v~ o ( {x_- 0--/ <~
0 <1-y-b/<S
C rt.- 0 < '/ _-6j-y
QS--___°'j_v _+_( _
j'_ z 2)
~ 1 (x/1) = L
~ J~) ➔ (°') ~)
~ ~ ;~ -~ lo ~ ~ DtL-
-5~~ ~ ~ ~kw-, 0<-/ '1) ~ L°'--~.
'P=reJie.i ~ ~ :
,0 ~ f(-x/ '1)± ~~ d( ,.__/ y) = J__ + 'YY L,
(){_ ,"i) ~ (G\ ,, b) (_-x., "1) -) (CA 0 I
0 \"'>=
~ -+ c~ /1) x ~➔ 3(_°'c'}6) 1- y_ Yvt ,
( 'X / '1) ~ l b)
(A ,I0( /1) j
_ k ~ f ('>t 1'i) = k~
0 J~k_f('»v
(2<- ,, '/) ➔ (o.};)
lX /i) ➔ (°' , ~
-f ('><- ,'i) J_
@ ~ fO
~ 1'f) ➔ (°'- ,b) B(~ / 1) - 'M - 1 l'V L
i°~ wi+lc, 'Yl.0
® -Jr re °""-J_ S ()Jr Q__
~ ~
~ ~ "'-- f'-darL ~ J s f d
p=vi U l% ~
j,: ,_ _ +C><:1) (Is = J_
%
' n 1
(c(x/'I_) ~ (<>-,, b) ) o.... ru..o.X___ ~ he re
- j O) + I <;, - -S-
~ L (?( -- 0 9(_
(-x,i)~(o /o) fi -J 'i
= !:0 ➔ (_o ,o)
L x{r,,- +rr)
= 0
, Jf-
= 0
LE½ 71Q)+·
~ S ho·w -tiJ ~ f( ~ /~ e_~ ~ -~ .
l!'-/'1) 7 (op)
1.,.,--
cSo /'D'l,:-
::; Let y = "YYL-, x_
1/m X 'j ~'VYl
-
-f (x_ /'1) = V
I+ rYL'l,-
·kry'
'V
+Y YYL'L- -
~ e_Q___. i_,m,, -f{~ ;'j) cLQ/) YlO+- ~xl/J+.
(_'l{ ;',) ➔ (9 /o)
Let /
