Transcript
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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
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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
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