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.

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00

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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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2006 Fall MATH 100 Lecture 8 3

MATH 100 Lecture 25 Final review

zyx

yx

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yx

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,,

f.Gradient

45) 15.4(Ex

.0& as 0,,

),(

aldifferenti Total e.

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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

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),( ima,maxima/min localint Unconstrah.

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ii

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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.

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E

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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

),,( ),,(

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integral Triple

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a

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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.

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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.

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integral line d.

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b

a

CC

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F

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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.

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f

x

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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

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1,,

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),(over ,function vector of integral Surface g.

r

zyx

R

yx

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hgfF

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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

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R

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