5vector function

8/6/2019 5Vector Function

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Vector Functions

Lesson 5

Objectives

At the end of the lesson you should be able to:

1. Define a vector function.

2. Find the limit of a vector function.

3. Differentiate a vector function.

4. Evaluate a given line integral.


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Vector FunctionsLesson 6

Objectives

At the end of the lesson you should beable to:

1. Define a vector function.

2. Find the limit of a vector function.

3. Differentiate a vector function.

4. Evaluate a given integral.


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Vector Function

.Rtandr(t)ofcomponentstheareh(t)andg(t),f(t),where

,h(t)g(t),f(t),r(t)

,kh(t)jg(t)if(t)r(t)

notation,In

vectors.ofsetaisrangewhoseandnumbersrealofset

aisdomainwhosefunctionaisfunctionA vector

"!

!

:Definition

ktj)tsin(i)t(cos(t)r2)

sint,t

e,t(t)r1):Examples2

!

"!


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Limit

The limit of a vector function r(t) is definedby taking limits of its component function,

That is ,

"

ppp

!

p

h(t)at

lim,g(t)at

limf(t),at

lim(t)rat

lim

provided the limits of of the componentfunctions exists.


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c ti sis(t)(t)f(t),fciflyifc ti sis(t)(t),f(t),r(t)f ctiv ct r

( ).r(t)rtliif,i t

tc ti sts iis(t)rf ctiv ct r

"!

!p

:Remark

:efiniti n

C ntinu us Vect r Functi n

ktj)tsin(i)t(cosr(t): !Exam les


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"! )(),(),()( thtgtftr

))(),(),((P thtgtf

x

z

o

Space Curve

.

acalledistandCofequationsparametriccalledareequationThese.

calledisI,intervalthethroughout

t variesandh(t)zg(t),yf(t),xwhere

space,inz)y,(x,pointsallofCsettheThenI.intervalanonfunctionvalued-real

continuousareh(t)andg(t)f(t),If

parameter

curveSpace

:Definition

!!!

y


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etc,.,(t)rrderivativeThird

(t)rrderivativeecond

h(t)rh)(tr

hlim

dtrd

(t)rdt

rd

functions.

valuedrealforaswaysametheindefined

is(t)rfunctionvectoraof(t)rderivativeThe

d

dd!ddd

dd!dd

p

!

d!

d

Derivative of a Vector Function


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10/24

f cti s.v lr lfsriv tivfr l sttsi il r

rf cti sv ct rfsriv tivf rl s

(t)(t),(t),f(t)rtf cti s,liff r tirf,r(t),(t)f(t),(t)rIf

:Remark

:Theorem

."ddd!d"!

Theorem


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Differentiation Rules

Conti

Rule).(Chain(f(t))u(t)f(f(t))]u[dt

d

6.

(t)v(t)u(t)v(t)u(t)]v(t)u[dt

d5.

(t)v(t)u(t)v(t)u(t)]v(t)u[

dt

d4.

dd!

dvvd!v

dd!


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Definition:

A curve given by a vector function r(t) on an interval Iis called smooth if r(t) is continuous and r(t) is not

equal to (except possibly at any endpoints of I).

Smooth Curves


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Integrals

The definite integral of a continuous vector r(t) can be

defined in the same way as for real valued functions

except that of the integral is a vector.

. !

!b

a

b

a

b

a

b

a

h(t) t)k(( (t) t)j(f(t) t)i(t

then,kh(t)j(t)if(t)r(t)If

r(t)

The Fundamental Theorem of Calculus:

R(a).R(b){R(t)}r(t) t thenr(t),(t)RIfb

a

ba ! !!d


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Arc Length and CurvatureLesson 7

At the end of the lesson you should be able to :

1. Definearc length.

2. Find arc length.3. Reparametrize the curve with respectto arc length.

4. Definecurvature.

5. Find curvature.

6. DefineTNB

7. FindTNB

8. DefineNormal plane and Osculating plane

9. Findthe Normal plane and Osculatingplane atpoint on a curve

UTP/JBJ 1


16/24

Arc Length

Definition:

!

!

!

!

ddd

ee!

b

a

b

a

dttrL

thtgtftr

kthjtgitftr

dtthtgtfL

ba

hg,f

btakthjtgitftr

)('Thus,

.)]('[)]('[)]('[)('havewe

,)(')(')(')('Since

.)]([)]('[)]('[

bygivenislengtharcitsthen,tofromonceexactly

traversediscurvetheIf.continuousareandwhere

,,)()()()(curvethattheSuppose

222

222

x

z

y

C


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Arc Length Function

Ifa curve is reparametrizedinterms ofits arc length(s)

thenitis convenientto understandthe curveinterms ofits

own arc length.

To obtaintheparameters (the arc length) we usethefollowing

arc length function:duurtss

t

a

)(')( !!

Differentiating bothsidesweget,

)( tr

dt

dsd!


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Curvature

Thecurvature ofa curveC (= ) at anygivenpoint

isa measure of how quickly the curve changes its direction at that

point. Moreprecisely, itisdefinedto bethe magnitude ofthe rate of

change ofthe unittangent vector with respectto arc length, andit

isdenoted by . Thus,

)(

)(

tr

tT

ds

Td!!O

)(tr

O


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Theorem

Theorem:

The curvature ofthe curvegiven bythe

vectorfunction is)(tr

3

)(

)()(

tr

trtr

d

ddvd!O


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Normal and Binormal Vectors

.Thus,

.

nt vector.unittangeais(t)Twhere

,

asde inedis

(t),rcurvespacesmoothaonpointgivenaAt

(t)(t)(t)

vectoritaalsoaaottolarer e ic

vectoraasefi eis(t)vectori ormale

(t)

(t)(t)

vector)ormalit

callesim ly(or(t)vectorormalU itPri ci alt e

v!

d

d!


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y

z

x

T

B

C

N

T - the unit Tangent vectorrepresents the

forward direction.

N - the unit Normal vectorrepresents the

direction in which your turning(when

you travel along the curve)

B - is the Binormal vector a tendency

of your motion to twistout of the plane

The T N B


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Normal and Osculating planes

Theplanedetermined bythe vectors and at a pointP

onCis calledthenormal plane.

Note:Thetangent vector becomesnormal vectorto thenormalplane.

Theplanedetermined bythe vectors and at a pointP

onCis calledtheosculating plane

Note:Thetangent vector becomesnormal vectorto thenormalplane.

)( trd

N B

N

B


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Osculating circle(or)

Circle of curvature

P.pointattheCofcurvaturetheiswhere

,

asdefinedisitandbydenoted

isPpointatthecircleosculatingtheofradiusThe

thecalledis

PatCofplaneosculatingintheliesthatcircleThe

O

1

curvatureofcirclethe(or)circleosculating

!


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Torsion

2

)(

bygivenisitandbydenotedisItP.pointaat

curveaoftwistingofdegreethemeasuresThe

rr

rrr

ddvd

dddddvd!T

T

torsio

5vector function

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