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2006 Fall MATH 100 Lecture 8 1
MATH 100 Lecture 25 Final review
44.)) 33, 15.2(Ex
),(),(lim
Continuity b.
existt doesn' ln
lim
),(lim
limits a.
00),(),(
22),(),(
),(),(
00
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yxfyxf
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Lyxf
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Class 25 Final review
1.Function of two or more variables
2006 Fall MATH 100 Lecture 8 2
MATH 100 Lecture 25 Final review
29) 11, 15.5(Ex
0)))
0),(),,(for
0)))
0
0 ofplant Tangent d.
40) 15.3(Ex
.' 0'22
1
ationdifferentiimplicit &function defined Implicitly c.
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z(zgy(ygx(xg
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g(x,y,z)
y
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yx
yx
zyx
2006 Fall MATH 100 Lecture 8 3
MATH 100 Lecture 25 Final review
zyx
yx
gggg
yx
yxyfxfyxf
,,
f.Gradient
45) 15.4(Ex
.0& as 0,,
),(
aldifferenti Total e.
121
21
2006 Fall MATH 100 Lecture 8 4
MATH 100 Lecture 25 Final review
D>0 Real min
Real max
D<0 Saddle saddle
D=0 ? ?
testpartial Second
),(0
0
),( ima,maxima/min localint Unconstrah.
19) 15.7Ex 30, 15.6(Ex
1 ,
sderivative lDirectiona g.
ii
u
yx
y
fx
f
yxfz
uuggD
0xxf 0xxf
(Ex 15.9 17 22)
2006 Fall MATH 100 Lecture 8 5
MATH 100 Lecture 25 Final review
17,22) 15.9x (
0),(
),(),(),(min
imamaxima/min constraintfor multiplier Lagrange i.
00
0000
),(
E
yxg
yxgyxfyxf
yxg
2006 Fall MATH 100 Lecture 8 6
MATH 100 Lecture 25 Final review
2. Integral
25) 8, 16.6Ex 17 11, 16.5Ex 31, 18, 16.2(Ex
),,( ),,(
}),( ),,(),(),,({
integral Triple
),(),(
} ),()(),({
integral Double a.
),(
),(
21
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)(
21
2
1
2
1
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b
a
xg
xgR
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dxdyyxfdAyxf
bxaxgyxgyxR
2006 Fall MATH 100 Lecture 8 7
MATH 100 Lecture 25 Final review
5) 17.5(Ex area. surface theyieldsit ,1when
1)),(,,(
),(over integral Surface c.
17) 16.8(Ex
),(
),()),(),,((),(
then
plane, in the into plane in the maps ),( and
),( if :Jacobians and integralsin variablesof Change b.
22
g
dAffyxfyxg
yxfz
dAvu
yxvuyvuxfdAyxf
xyRuvSvuyy
vuxx
R
xx
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uv
R
xy
2006 Fall MATH 100 Lecture 8 8
MATH 100 Lecture 25 Final review
18) 14, 17.3Ex ( 0 then ),,(),( if ,particularIn
),(),(
and function,
potential its is and field vector veconservati a called is
, If :functions potential and fieldon Conservati e.
41) 27, 17.2(Ex
))(')(),(),(())(')(),(),(())(')(),(),((
integral line d.
0011
0011
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C
b
a
CC
rdFyxyx
yxyxrdF
F
F
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hdzgdyfdxrdF
2006 Fall MATH 100 Lecture 8 9
MATH 100 Lecture 25 Final review
11,23) 17.4(Ex 0 then ve,conservati is If
.
thenckwise,counterclo oriented is which curveby
enclosed region aover continuous are sderivativeorder
1st their and , If .,Let : theoremsGreen' f.
C
RC
rdFF
dAy
f
x
grdF
C
R
gfgfF
2006 Fall MATH 100 Lecture 8 10
MATH 100 Lecture 25 Final review
14) 9, 17.8(Ex
)(
normal outward surface, closed
,,, : theoremDivergence h.
19) 9, (Ex17.7 ly.respective
,orientaion upward"" and downard"" tocorrespond "-" and ""
1,,
norientatiowith
),(over ,function vector of integral Surface g.
r
zyx
R
yx
dVhgfdSnF
hgfF
dAzzFdSnF
yxfzF
2006 Fall MATH 100 Lecture 8 11
MATH 100 Lecture 25 Final review
3. Integration with other coordinate
16) 13, 16.7Ex (
sin),,(
scoordinate Spherical b.
,),,(
),(
} ),()(),({ coordinatePolar a.
2
1
2
1
2
1
2
1
2
1
2
1
2
),(
),(
)(
)(
2121
dddffdV
rdrddzzrffdV
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V
R
rg
rgV
r
rR
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