Vector Functions 9-1

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Vector FunctionsThis unit is based on Section 9.1, Chapter 9, of the textbook. All assigned readings and exercises are from the textbookObjectives:Make certain that you can define, and use in context, the terms,

concepts and formulas listed below:1. scalar and vector functions and fields2. differentiate a vector function 3. calculate the second derivative and the integral of a vector

function 4. find the limit (if it exists) of a vector function as t→ to.5. sketch the curve traced by a vector function and identify the

vector function that traces a given curve.6. identify the vector function , given its derivative .7. apply operations of vector algebra to vector functions to form

new vectors8. determine the length of a given curve.9. define a vector tangent to a given curve at a given point.10.solve practical problems

Reading: Read Section 9.1, pages 452-457.Exercises: Complete problems

)(trr

)(trr

)(trr

)(trr )(' trr

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PrerequisitesBefore starting this Section you should . . .1. be familiar with the concept of vectors2. be familiar with vector algebra3. be familiar with scalar functions

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Vector Calculus: Scalar Fields and Vector FieldsEngineers use vector calculus to define and measure the variation of temperature, fluid velocity, force, magnetic flux etc. over all three dimensions of space. In the real 3D engineering world, one wants to know things like the stress and strain inside a structure, the velocity of the air flow over a wing, or the induced electromagnetic field inside a human body.A scalar field in a given region of 3D space is a scalar function defined at each point in the region, i.e. f(x,y,z). Examples: electric potential, gravitational potential, …A vector field in a given region of 3D space is a vector function defined at each point in the region, v(x,y,z). Examples: electric force field, gravitational force field, …A field may also depend on time, i.e., temperature inside a roomf(x,y,z,t) or fluid velocity v(x,y,z,t). We should know how to integrate and differentiate vector quantities with three components which depend on three co-ordinates x; y; z.

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Visualization of scalar and vector fields

V(x,y,z,t)

f(x,y,z,t)

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Scalar and vector fields in 3D.

Consider a metallic plate that is heated on one side and cooled on another.

The temperature at each point within the body is described by a scalar function (field) T(x,y,z,t). The flow of a heat may be marked by a filed of arrows indicating the direction and magnitude of flow. This energy or heat flux is described by a vector function (field) H(x,y,z,t).

hot cold

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9.1 Vector Functions

ktzyxFjtzyxFitzyxFtzyxF zyxˆ),,,(ˆ),,,(ˆ),,,(),,,( ++=

r

A vector function is a vector that each component is a function of one or multiple variables.

Functions of several variables: examples• Electric and magnetic field as a function of position

and/or time,• Heat flux as a function of position and/or time• Force on a particle as function of position and time

kzyxFjzyxFizyxFzyxF zyxˆ),,(ˆ),,(ˆ),,(),,( ++=

rStatic force:

Time varying force:

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Vector Functions

In this section the focus will be on vector functions of single variable for studying the motion on a curve.

Figure 1

Functions of single variable:q(t) = f(t)i + g(t)j + h(t)k

Physical Examples: Position of a particle versus time in 2D Space

r(t) = x(t) i + y(t) j (Figure 1(a))Position of a particle versus time in 3D-Space

r(t) = x(t) i + y(t) j + z(t) k (Figure 1(b))

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Limit of Vector Functions

Definition 9.1 Limit of a Vector Function

.)(lim),(lim),(lim)(lim

)(lim)(lim),(lim

>=<→→→→

→→→

thtgtft

thtgtf

atatatat

atatat

r

then exits, and If

then,)(lim and )(lim If 22at11attt LrLr ==

→→

2121

2121

11

ttiii

tt

cctci

LLrr

LLrr

Lr

⋅=⋅

+=+

=

→

→

→

)]()([lim )(

)]()([lim (ii)

scalar a ,)(lim )(

at

at

at

>=< )(),(),()( thtgtftr Let

Theorem 9.1 Properties of Limits

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Example: Graph the curve traced by the vector

jeietr tt ˆˆ)( 2+=r

Step 1: write parametric equations for the curve:

x(t) = et and y(t) = e2t

Step 2: construct a table using different values of t and corresponding x and y values:

......

....2

....1

....0yxt

Step 3: plot y values vs x values

x

y

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Example: Find the vector function that describes the curve of intersection of the surfaces:

xyyxz =+= and 22

Step 1: set x = t

Step 2: sub for x in the equations

∴ y = t and z = t2 + t2

Step 3: write the vector form of the curve

ktjtitzyxr ˆ2ˆˆ,, 2++>==<r

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Continuity of a Vector Function

Definition 9.2 Continuity A vector function r is said to be continuous at t=a if

Equivalently, r(t) is continuous at t=a if and only if f(t), g(t), and h(t) are continuous there.

).()( lim (iii) and exists, )( lim (ii) , defined is )( )(atat

attai rrrr =→→

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>′′′=<′>=<

)(),(),()(,,,)(),(),()(

thtgtfthgfthtgtft

r then able,differenti are and where r If

Differentiation of vectorsConsider a vector r(t) that is a function of a scalar parameter (variable) t. The derivative of r(t) with respect to t is defined as

ttrttr

dtrd

t ∆−∆+

=→∆

)()(lim0

rrr

Note that r′ (t) is also a vector, which is not parallel to r(t).

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Higher Order Derivative Second order

)(),(),()( >′′′′′′=<′′ thtgtftr

Example: Find >−=< ttttetrtr t 232 4,,)(given )(' rr

First differentiate each component and then write as a vector:

ktjtieter tt ˆ)18(ˆ3ˆ)2(' 222 −+++=r

>=<>−=<>=<=

++++=

8,0,4",1,0,1',0,0,0

ˆ8ˆ6ˆ]2)12(2[)(" 22

rrr

kjtieettr tt

rrr

r

& 0, t at Note

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Geometric Interpretation

)()∆(∆ ttt rrr −+=

)]()∆([∆∆

∆ tttt

1t

rrr−+=

)(tr′ is tangent to C at P

0r ≠′ )(t at P

Smooth Curve (Smooth Function)

A function is called smooth function and the curve traced is called smooth curve, if

1) Components of r have continuous first derivative2) r′ (t) ≠ 0 over the open interval (a , b)

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Example Tangent Vectors

/6).(r and (0)r Graph B) .j i r

by given is positionwhosepointa bytracedisthatcurve the Graph A)

πrr

r

′′≤≤+= πt, ttt

PC

20ˆsinˆ2cos)(

11,21sin,2cos

2 ≤≤−−=∴

==

xyxtytx with

j i r ttt cos2sin2)( +−=′∴

jir and jr233)6/()0( +−=′=′ π

Solution: from r expression

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Rules of Differentiation & Chain RuleAssuming r1 and r2 are differentiable vector functions of a scalar t, and f is a differentiable scalar function of t:

[ ]

[ ]

[ ]

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

)()(')(')()()(

)()(')(')()()(

)(')(')()(

212121

212121

2121

trtrtrtrtrtrdtd

trtrtrtrtrtrdtd

trtftrtftrtfdtd

trtrtrtrdtd

rrrrrro

rrrrrro

rrro

rrrro

×+×=×

•+•=•

+=

+=+

)(')(')( tssrdtds

dsrd

dtsrd r

rro ==

If a vector r(s) is a function of the scalar variable s, which is itself a function of t such that s = s(t), then we have

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Example: Chain Rule

[ ]kji

k j i r

rrthen

and k j i r If

4334343

33

4

3

12)2cos(8)2sin(8

43)2cos(2)2sin(2

,,)2sin()2cos()(

t

s

s

ettttt

tessdtd

dtds

dsd

dtd

tsesss

−

−

−

−+−=

−+−=

=

=

++=

o

o

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Integration of vectorsConsider a vector

that is a function of a scalar variable t. The integral of r(t) with

respect to t is defined as

∫∫∫∫ ++= dtthkdttgjdttfidttr )(ˆ)(ˆ)(ˆ)(r

kthjtgitftr ˆ)(ˆ)(ˆ)()( ++=r

Example: Problem 9.1-33

kji

dttkdttjtdti

dtktjtit

ˆ15ˆ9ˆ5.1

4ˆ3ˆˆ

)ˆ4ˆ3ˆ(2

1

32

1

22

1

2

1

32

++=

++=

++

∫∫∫∫

−−−

−

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Application: Length of a curve

∫=2

1

|)('|t

tdttrs r

a

bThe arc length (s) between two points:

t = t1 (point a) and t = t2 (point b),

on a space curve r(t), is given by

Example:

.............2

2|'|

cossin,sincos,1)('0,sin,cos,)(

0

2

2

=+=∴

+=

>+−=<≤≤>=<

∫π

π

dtts

tr

tttttttrtttttttr

r

r

r

Then Given

CORRECTION