Math 474 : Dy

÷in They

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:

Def: A smooth paramelvizd come in R3 LN R'

,or gennlf R

" ) is a smooth ( i.e. , infinitely differentiable) mp a .la ,b) - R3

bhe a ,b ER .

we using write altklxlttylttzttl ) & the velocity vectr ar tart veetr B a'

HKIXH, y

' At,Eta )

.

by : HH =L a cost,

a sntibt )

µ ht : The aknay It, Cashed )

y

a

The image of alt ) is a helix I ✓±.

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3 2

at : the cubic xH= It,t )

*We asnlf art to exclude this bstexaple by assay regularity

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Def : A smooth partied one alt ) B regular if a' IH to for all t in the domain

.

Df: The arc length f a onthnoif difndyle ( C' ) one alt ) blw to & t . 3 defind to be

sktftoldlulldu

If I dlu ) 1=1 t n,then we oboe tht SH = t - to & usgx is poametited by wdegth .

It turns at to be theaetalfomnint to assume any are pretrial by adyth,so we'll usnlfmhethtassnptmahn

pay themes . Homer,its aguttmf iprdicd to get cares to be paraded by rdeyth, so we'll also need A be able to

Compute wl other pranetrizeths .w,the lgth f a C

'

one a :[ a. 4) → "23 is it |gthH= SHKS! la' HH at

.

pat It if x say ats? lsgitb arms ) Dos it still health ?

Intrstjf, the sur is : sometimes .

It it A

It itDFor a anti his one a : Ca , 5) → 1123

, we define :.

atnart.kmjyipgI.E.taiiI.onIy.at@adnebfie1eythkt-snpplkiD.DfiTfleglhHBfnitgnesyaBareetfnbleame.t

# : The Kah snwfbhe B the limit of the flkg que f guts :

:-, m- m

. ..

0 1 / z2 / z I

i. c .

,at eh Ap name the middle t.nl of mhntl & rphe it wlte other to say f an gniktl trifle

based at tht sbsgnt .

Its rt so hd to an vine ymsdf tut the Koh swµ B a ants are Hfhobvif mt dilated,ht

the His he tftb 1413T =L,413

,

141312,

.. .

.

Sine hh is an insulted polyp in the ft one,he

got a same f prttis Pi st . lla,Pit → x

, mj lythk) des at exist & a is nt rettdle.

Thin : Tf a s d,

Fm or to nots f lythk) agree

.