Download - 2. Vector Calculus
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VECTOR CALCULUS
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Vector Product
a
b abCROSS PRODUCT
multiplication VECTOR
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Vector Product (contd.)
a
b
Magnitude :Area of the parallelogram
generatedby a and b.
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Vector Product (contd.)
a
b
Magnitude: sinabah
sinbh
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Vector Product (contd.)
a
b
ba
sinbh
Direction : Perpendicular to
both a and b.
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Vector Product (contd.)
a
b
ba
Direction : A rule is required !!
sinbh
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a b
ba
Right-Hand Rule
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The order of vector multiplicationis important.
a
b
ab
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Geometrical Interpretation
a
b
bsin
sin|ba| ab
A =a b
Area of the parallelogram
formed by a and b
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Properties
Vector multiplication is notCommutative.
Vector multiplication is Distributive
caba)cb(a
Multiplication by a scalar
mmmm )ba()b(ab)a()ba(
abba
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Properties(contd.)
00sin|ba| ab
and aand bare notnull vectors, then ais
parallel to b.
If 0ba
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Properties(contd.)
jik;ikj;kji
Angle between them 0
0k
k
j
j
i
i
Angle between them =90
jki;ijk;kij
j
i
k
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Vector Product: Components
a
b
)aa(a k
3j
2i
1 )bb(b k
3j
2i
1
)k
3j
2i
1(k
3
)k3
j2
i1
(j2
)k3
j2
i1
(i1
bbba
bbba
bbba
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Vector Product: Components
)kk(
33
)jk(
23
)ik(
13
)kj(32
)jj(22
)ij(12
)ki(31
)ji(21
)ii(11
bababa
bababa
bababa
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Vector Product: Components
k)1221
(
j)3113
(i)2332
(
ba
baba
babababa
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Vector Product: Determinant
321
321
kji
bbb
aaaba
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Examples in Physics
z
x
yF
r
Fr
The torque
produced by
a force is
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x
y
z
O
The angularmomentum
of a particle
with respectto O
Examples in Physics (contd)
vp
mr
prL
prL
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Examples in Physics (contd)
The force actingon a charged
particle moving
in a magneticfield,
)Bv(F
qv
B
F
F
Positive charge
Negative charge
l
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Applications
)32()25( kjikji A simple cross product
132
215
kji
k
j
i
)]1.2(3.5[
)]1.5(2.2[
]2.3)1.1[(
17
9
-5
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kji 1795
C
H
EC
K
)( ba
is perpendicular to ba
&
An Example
)
3
2()
2
5( kjikji
017.29.15.5
)
17
9
5).(
2
5(
kjikji
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finding a unit vectorperpendicular to a plane.
F ind a unit vector perpendicular
to the plane containing two vectors b&a
Applications(contd.)
A vector perpendicular to a
and b is
Corresponding unit vectorba
|ba|
ba
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kjiba 1795
Cross product
Magnitude 3952898125|| ba
Unit vector is )1795(395
1kji
An ExampleDetermine a unit vector perpendicular
to the plane of
and
)25( kjia
)32( kjib
SUMMARY
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SUMMARY
Magnitude and Direction of a vectorremain invariant under transformation
of coordinates.
Product of a vector with a scalar is a
vector quantity
Vector product : directional property,
denotes an area.
Scalar product
a. b =axbx+ayby+ azbz
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TRIPLE PRODUCTS
Scalar Triple Product
)cb(a
Vector Triple Product
)( cba
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Scalar Triple Product
)]3
2
1()
3
2
1[(
)3
2
1(
kcjcickbjbib
kajaia
])1221
(
)3113(
)2332[(
)3
2
1(
kcbcb
jcbcbicbcb
kajaia
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Scalar Triple Product (contd.)
)()()(
12213
3113223321
cbcbacbcbacbcba
321
321
321
ccc
bbb
aaa
)(
cba
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Scalar Triple Product (contd.)
a
b
c
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sin|| bccb
b
c
bsin
|| cb
Area of the base
Scalar Triple Product (contd.)cb
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a
b
c
cb
acos = height
cos|||)(| cbacba
Volume of the parallelopiped
Scalar Triple Product (contd.)
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Properties
Interchanging any two rows reversesthe sign of the determinant, so
)bc(a)cb(a
Interchanging rows twice the original
sign is restored, so
)ba(c)ac(b)cb(a
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Properties
If any two vectors of the scalar tripleproduct are equal, the scalar triple
product is zero.
0)( caa
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bac - cabrule
Vector Triple Product
)( cba
).().( baccab
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SUMMARY
A physical quantity which hasboth a magnitude and a direction
is represented by a vector
A geometrical representation
An analytical description: components
Can be resolved into components along
any three directions which are non
planar.
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SUMMARY
Vector Product
k)1221
(
j)3113
(i)2332
(ba
baba
babababa
Scalar Product of vectors
a. b =a1b1+a2b2+ a3b3
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SUMMARY
Scalar Triple Product : volume of aparallelepiped.
)cb(a
Vector Triple Product
Quadruple Product of vectors
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z
fk
y
fj
x
fizyxfF
),,(
zk
yj
xi
F
i.e. gradient of a scalar quantity is a vector quantity,
),,( zyxf is a scalar quantity
Geometrical Interpretation:Gradient has magnitude and direction
( , , ). cosdf f x y z dl f dl
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For fix value of magnitude of dl, df is greatestwhen cos is zero, i.e. we move in the samedirection as f.oThe gradient f points in the direction ofmaximum increase of the function f.
oThe magnitude f gives the slope (rate ofincrease) along this maximal direction.
( , , ). cosdf f x y z dl f dl
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DIVERGENCE
z
F
y
F
x
F
Fzyx
.
is a vector quantity.
is a scalar quantity
F
F
.
F
. is known as divergence of a vector quantity ( )
Physical Significance
It represents how much the vector spreads out (diverges) from the point. If divergence of any
vector is positive then it shows Spreading out
and if negative then coming towards that point.
F
zk
yj
xi
x y zF iF jF kF
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Eg: Divergence of current density:
(Current density : current per unit area)
at a point gives the amount of charge flowing
out per second per unit volume from a small
closed surface surrounding the point.
j
( ) 0div v i.e. the flux entering any element of spaceis exactly balanced by that leaving it.
Such vectors are known as solenoidal vector
Point works as Source Point works as Sink
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CURL
is a vector quantity
is a vector quantity known as curl of
Physical Significance
It is a measure of how much the vector curls
around the point.
zyx kFjFiF
z
k
y
j
x
iF
F
F
F
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PROBLEMS
1. If A=3x2y - y3x2, calculate gradient A at a point
(1,-2,-1) 2. If = x2yi-2xzj+2yzk, calculate divergence and
curl of a vector at (1,2,1).
Ans: 1. 10i-9j
2.(i) 6 (ii )k
A
A
SECOND DERIVATIVES
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SECOND DERIVATIVES
The gradient, the divergence and the curl are the only
first derivatives we can make with , by applying
twice we can construct five species of secondderivatives.
The gradient is a vector, so we can take the divergence
and curl of it.
(1) Divergence of gradient : (Laplacian)(2) Curl of gradient:
o The divergence is a scalar, so we can take its gradient.
(3) Gradient of divergence.
o The curl is a vector, so we can take its divergence and
curl.
(4) Divergence of a Curl.
0)( A
).( A
0).( A
AA 2).(