2 scalar and vector field
DESCRIPTION
brief description of scalar and vector fields required for mathematical analysisTRANSCRIPT
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Scalar and Vector FieldsScalar Field :
A scalar quantity, smoothly assigned to each point of a certain region of space is called a scalar fieldExamples :i) Temperature and pressure
distribution in the atmosphereii) Gravitational potential around the
earth
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iii) Assignment to each point, its distance from a fixed point
222 zyxr
O
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O
),,( zyx
),,( zyxf
x
z
y
Once a coordinate system is set up, a scalar field is mathematically represented by a function : )(),,( rfzyxf
is the value of the scalar assigned to the point (x,y,z)
),,( zyxf
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A smooth scalar field implies that the function , is a smooth or differentiable function of its arguments, x,y,z.
),,( zyxf
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Since the scalar field has a definite value at each point, we must have
),,(),,( zyxfzyxf
O),,( zyxf
Consider two coordinate systems.
x
y
z
O’
z
y
x
),,( zyxf
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Vector Fields :
A vector quantity, smoothly assigned to each point in a certain region of space is called a vector field
Examples :
i) Electric field around a charged bodyii) Velocity variation within a steady flow of fluid
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iii) Position vector assigned to each point
Or
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Once a coordinate system is fixed, a vector field is mathematically represented by a vector function of position coordinates : )(),,( rForzyxF
O
)(rF
r
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Resolving the vector at each point into its three components, the vector field can be written as :
kzyxFjzyxFizyxFzyxF zyxˆ),,(ˆ),,(ˆ),,(),,(
A smooth vector field implies that the three functions, , are smooth or differentiable functions of the three coordinates x,y,z.
zyx FFF &,
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),(sin),(cos),( yxFyxFyxF yxx
),(cos),(sin),( yxFyxFyxF yxy
x
yF
x
y
How Are the Component Functions in Two Frames Related to One another?
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In Three Dimensions :
zyxx FRFRFRF 131211
zyxy FRFRFRF 232221
zyxz FRFRFRF 333231
MatrixRotationRRRRRRRRR
R
333231
232221
131211
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Gradient of a Scalar Field
dzzfdy
yfdx
xfrfrdrfdf
)()(
O
r rdr
rd
kdzjdyidxkzfj
yfi
xf ˆˆˆˆˆˆ
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rdf
Where, we have the shorthand notation :
fkzfj
yfi
xf
ˆˆˆ
Since is a vector assigned to each point
f
it defines a vector field. This vector field is called the gradient of the scalar field
f
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We have
df )ˆ(ˆ ndsrddsnfrdf
nfdsdf
n
ˆˆ
That is, the rate of change of a scalar field in any direction at a point, is the component of the gradient of the field in the given direction, at that point.
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Thus, gradient of a scalar field at any point may be defined as a vector, whose direction is the direction in which the scalar increases most rapidly, and whose magnitude is the maximum rate of change
fdsdfnf
dsdf
n
maxˆ
ˆ
And the maximum value occurs when is in the same direction as
n̂f
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Ex. 1.3Find the gradient of the scalar field :
222)( zyxrrf
and show that it has the properties as stated.
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Prob. 1.12The height of a certain hill (in feet) is given by :
)122818432(10),( 22 yxyxxyyxh
where x & y are distances (in miles) measured along two mutually perpendicular directions from a certain point. (c) How steep is the hill at the place (1 mi,
1 mi)(d) In which direction, must one move at this point so that the slope is 220 ft/mi
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x
y
),( yxh
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x
y
),( yxh
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Further Examples :
)()( rFrf b)
Let Then )()() rrFrrfi
)()() rrFrrfii
gdgdfrgfa
))(()
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Prob. 1.13
Let r be the separation vector from a fixed point to the point and let r be its length. Find :
),,( zyx ),,( zyx
a) (r2)
b) (1/r)
c) (rn)
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The ‘Del’ Operator
The gradient of a scalar field can be thought of as the result of the vector differential operator :
zk
yj
xi
ˆˆˆ
acting on the scalar field :f
zk
yj
xif
ˆˆˆ
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The gradient (2-dim) will be a vector field, if, at every point in space, we have :
yf
xf
xf
sincos
yf
xf
yf
cossin
Is the Gradient of a Scalar Field a Vector Field?
Prob. 1.14
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Important Properties of the Gradient1. The line integral of the gradient of a scalar field from one point to another, is independent of the path of integration (it depends only on the two points)
)()( 12
21
PfPfldfldfCC
1C2C
1P
2P
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2. Line integral of the gradient around a closed path is zero
0ldf
3. If the line integral of a vector field from one point to another is independent of the path joining them, then that vector field must be the gradient of a scalar field.
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),,( zyx
:),,( zyxffieldscalartheConstruct
),,(
)0,0,0(
),,(zyx
ldFzyxf
),,(int zyxpothetoorigintheconnectingpatharbitraryanyispaththewhere
Proof :Let be a vector field whose line integral is path independent.
F
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x y z
zyx zdzyxFydyxFxdxFzyxf0 0 0
),,()0,,()0,0,(),,(
),,( zyx
)0,0,(x)0,,( yx
Choosing the path as above
To show : Ff
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),,( zyxFzf
z
Similarly, choosing the paths as below,
),,( zyx
)0,0,(x
),0,( zx ),,( zyx
),,0( zy),0,0( z
xy FxfF
yf
&
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fF
),,( zyx
),,( 000 zyx
),,(
),,( 000
),,(zyx
zyx
ldFzyxg
However, the scalar field, whose gradient is the vector field , is not unique.
F
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We still have : gF
),,(
)0,0,0(
000
),,(),,(zyx
ldFzyxgzyxf
Czyxg ),,(
However,
That is, the two scalar fields, which give us the same vector field as their gradient, differ from each other by a constant.
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If the scalar field is constructed as :
),,(
),,( 000
),,(zyx
zyx
ldFzyxf
then :0),,( 000 zyxf
Since are arbitrary, the zero of the scalar field is arbitrary.
000 &, zyx
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Level or constant surfaces of a scalar field
Given a scalar field , the locus of all points which
satisfy the condition :
),,( zyxf
Czyxf ),,(
defines a surface, known as the level or constant surface
of the field
Czyxf ),,(
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1C2C
3C
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),,(),,( 000 zyxfzyxf
Through every point in space, one can draw a level surface of :
),,( 000 zyxf
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Level surfaces of the scalar field :222),,( zyxzyxf
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4. The gradient of a scalar field, at each point in space, is perpendicular to the level surface (constant surface) of the scalar field, passing through that point
.),,( Constzyxf
Pf
P
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Ex.: Find the unit normal to the surface :
22 yxz
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Given the three components of a vector field, , one can construct nine first derivatives :
zyx FFF &,
zF
yF
xF zxx
.,..........,,
What scalar and vector fields can be constructed out of these nine first derivatives?
Divergence and Curl of a Vector Field
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Correction!
x
y
'x
'y
),( 00 yx
Coordinate Transformation Equations
;)(sin)(cos 00 yyxxx
)(cos)(sin 00 yyxxy
;sincos 0xyxx 0cossin yyxy
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It turns out that only one scalar field and one vector field can be constructed out of these nine first derivatives :Divergence :
zF
yF
xFF zyx
Curl :
kyF
xF
jxF
zFi
zF
yFF xyzxyz ˆˆˆ
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kFjFiFz
ky
jx
iF zyxˆˆˆˆˆˆ
kFjFiFz
ky
jx
iF zyxˆˆˆˆˆˆ
zyx FFFzyx
kji
ˆˆˆ
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Example :
Let the vector field be : rrF
)(
3 F
0
F
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Surface Integral of a Vector Field
F
ad
S
S
adF
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i) Gauss’ Divergence Theorem :
S V
dFadF
VS
Here, is any vector field and , any closed surface
F
S
Two Important Theorems
ad
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A Simple Application of Divergence Theorem
General Proof of Archimedes’ Principle
S S
b adPFdF
xy
zFd dah )()( zhgzP
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ii) Stokes’ Curl Theorem
ld
ad
C S
adFldF )(
Here, C is any closed loop (planar or otherwise), and S is any surface bounded by the loop.
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Second DerivativesfieldVectorf
2
2
2
2
2
2
)(zf
yf
xff
i) f2
Where is a second derivative operator, known as the Laplacian.
2
2
2
2
2
22
zyx
ii) 0
f
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fieldscalarAF
F
a legitimate vector field, which is not much used.
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fieldvectorAF
0) Fi
FFFii
2)
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Certain Product Rules
)()()(.1 fggffg
)()()(.2 ABBABA
ABBA)()(
)()()(.3 fFFfFf
)()()(.4 BAABBA
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Prob. 1.60Show that
V S
adTdTa
)()(
Hence show that :0
S
adaProb. 1.61b :
Prob. 1.61c : a is the same for all surfaces sharing the same boundary.
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Prob. 1.61a : Find the vector area of a hemispherical bowlBack to Prob. 1.60
V S
adFdFb )()(
V S
adTUUTdTUUTd
)()()( 22
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Prob. 1.61d
Show that : ldra
21
Proof :
C S
adrAldrA )]([)(
Use the product rule :BAABBA)()()(
)()( ABBA
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Two Important Results
i) If the Curl of a vector field vanishes, then that vector field must be the gradient of a scalar field :
fFtsfF
..0
We have seen : 0)0) Fiifi
The converse results are also true.
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ii) If the Divergence of a vector field vanishes, then the vector field must be the Curl of a scalar field :
GFtsGF
..,0
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Theorem I : The following statements are equivalent.(A) Vector field is such that its line integral around any closed path is zero
F
(C) Vector field is the gradient of a scalar field
F
(B) Vector field is such that its line integral from one point to another is independent of the path joining the two.
F
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)()()()( DCBA Or
)()()()()( ADCBA
(D) Vector field is such that it has a vanishing curl
F