2. Vector Calculus

8/22/2019 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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a

b

Magnitude: sinabah

sinbh


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a

b

ba

sinbh

Direction : Perpendicular to

both a and b.


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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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)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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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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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



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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


)( 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


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).(

2. Vector Calculus

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