Vector Analysis

Electromagnetic Theory

PHYS 401

Fall 2017

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Vector AnalysisVector analysis is a mathematical formalism with which EM concepts are most conveniently expressed and best comprehended.

• Many physical quantities are completely describes by their value

– e.g. temperature, pressure, mass, frequency…

• Such quantities are called scalars, and their values can be given in numbers.

• But many physical quantities have a direction in addition to magnitude– e.g.: velocity, force, displacement…

– To describe such quantities their direction as well as their magnitudes have to be specified. So just a regular number is not inadequate.

Vector: a mathematical object that has a magnitude and a direction.

– Physical quantities which possess a direction as well as a

magnitude are represented by vectors

Vector Notation: A vector is usually written as a bold face letter (like A) or with an little arrow or line above it �� , �̅

The magnitude of a vector A is written as |A| or A2

• Often a vector is graphically represented by a directed line segment (an arrow):

– Whose length represents the magnitude and its orientation in the direction of the vector.

NB.: This is just a geometrical representation, vectors are not arrows, they are abstract mathematical objects.

In representing a vector graphically, it is defined only by its magnitude and direction, not its position.

length = |A|, A

magnitude of vector

direction:

direction of vector

All these represent the same vector

A

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• When a vector is multiplied by a positive number, it multiplies its magnitude, its direction stays the same.

• Multiplying by a negative number flips the direction of a vector

A 2A2.5A

-2A

• Sum of two vectors is the single equivalent vector which has same effect as application of the two vectors.

• e.g. consider adding two displacements.

A

BA+BGraphically, vectors are added using the

triangular rule

(also called “head-to-tail” rule; Move B so that the head of A touches the tail of B)

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• This can be repeated to add any number of vectors

AB

A+BCA+B+C

given any three vectors A, B, and C vector addition obey following properties:

Addition Multiplication by a scaler

Commutative

Associative

Distributive

where k and l are scalars

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• Just like vectors can be added to form a single vector, any given vector can be written as a combination of other vectors.

• This is called vector decomposition.

• One particularly useful decomposition is, decomposing a vector as a sum of vectors parallel to coordinate axes.

A

X

Y

X

Y

A

X

Y

Y

X

Z

Z

A=X+YA=X+Y+Z

• Those are called components of the vector along coordinate axes

Vector Decomposition

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• Unit vectors: A vector whose magnitude is 1.

– Usually unit vectors are written with a hat (like �� , �� =1 )

– Any vector can be written as a product of its magnitude times the unit vector in that direction.

��

aa= � ��

�� : the unit vector in the direction of the vector a

X

A

X

Y

Y

Z

Z

A=X+Y+Z

��

��

Suppose �� , �� , are unit vectors along X,Y,Z coordinate directions,

Then X=|X| �� , let |X| = x then X= x��

Similarly Y= y�� , Z= z

So A=X+Y+Z = x��+ y�� + z

Often it is just written as a coordinate triplet

A= (x,y,z) leaving the sum over unit vectors to

be understood.

Magnitude of the vector A= � � � � �

⇒ �� = �

�

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Position Vector (Radius Vector)

The position vector r of point P is defined as the directed distance from the origin O to P; that is,

r = x��+ y�� + z

• The unit vector in the direction of r is

� =�

�=

��� � ��� � �

� � � � �

• If there are two points with position vectors r1 and r2 , thedistance vector between them r12 = r2 – r1

X

r

xy

Y

Z

z

P

��

��

O

8

O

r1

r2

P1

P2

r12

• Components of sum of vectors is sum of components

e.g. Let A = ����+ ���� + �� and B = ����+ ���� + ��

Then A+B = �����+ ���� + �� �� ) +(����+ ���� + �� )

= ����+���� +���� +����+ �� + ��

= ���������+���������+���+���

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Vector Products• Vector quantities are often combined to form a new quantities.

the result could be a scale or a vector.

– e.g. Work done by a force is the product of displacement times the force in the direction of displacement.

• Both force and displacement are vector quantities, and work is a scalar.

– Angular momentum is the product of momentum and distance perpendicular to momentum to origin (axis of rotation).

• Momentum is a vector, since distance is taken in a specified direction it also becomes a vector. Resulting angular momentum is also a vector.

• So two types of vector products are defined, scalar and vector.

� = �� cos "

p

rO

"

��

# = ��$% sin "

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The Scalar Product (Dot Product)

The scalar product (also called of the two vectors A and B is defined as the product of the magnitude of A and the projection of B onto A (or vice versa):

( · * = AB cos "-.

where "-. is the angle between A and B.

dot product is commutative : ( · * = * · (

and distributive : ( · * + / = ( · * + ( · /

• If two vectors are perpendicular to each other their scaler product is zero (cos90∘ = 0) .

• Therefore for unit vectors �� , ��, along coordinate axes

�� · �� = 4; �� · = 4; �� · = 4

and �� · �� = 5; �� · �� = 5; · = 5

"-.A

B

��

��11

if A = ����+ ���� + �� and B = ����+ ���� + ��

( · * = �����+ ���� + �� � · ����� � ���� � �� �

using above properties = ����+ ����+ ����

( · ( = ����+ ����+ ���� = ��

� ��

���

= (

Since ( · * = AB cos "-.

cos "-. =(·*

( *useful in finding angle

between two vectors.

In general the component of a vector ( in the direction of vector C is given by ( · /7

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The Vector Product(Cross Product)

• The vector product of two vectors, A and B is a vector,

–its magnitude is equal to the product of the magnitudes of A and B and the sine of the angle between them

–its direction is perpendicular to the plane containing A and B, in the direction of advance of a right handed screw when it is turned from A to B.

( 8 * = 9�AB sin "-.

( 8 * = area of the parallelogram determined by A and B

A

B

"-.

:�

( 8 * = 9�AB sin "-.

( 8 *

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• The vector product has following properties:

– It is not commutative: ( 8 *= ;* 8 (

– It is not associative: ( 8 * 8 / < ( 8 * 8 /

– It is distributive: ( 8 * � / = ( 8 * � ( 8 /

– ( 8 ( = 4�sin θ = 0�

• For unit vectors �� , ��, along coordinate axes

�� 8 �� = 4; �� 8 �� = 4; 8 = 4

�� 8 �� = ; �� 8 = ��; 8 �� = ��

�� 8 �� = ; ... and so on

In tems of components

A = ����+ ���� + �� and B = ����+ ���� + ��

( 8 * = �����+ ���� + �� � 8 ���� � ���� � ��

= ����� ; ������� � ����� ; ������� � ����� ; �����

��

��

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Triple vector products

• Since result of the vector product between two vectors gives a vector it can it can be multipied with another vector.

• The scalar triple product between A ,B and C :

– It is equal to the volume of the parallelepiped spanned by A, B, and C

In terms of components

• The vector triple product between A ,B and C :

( 8 * 8 / = * ( · / ; /�( · *� (“bac-cab” rule)

It is not associative: ( 8 * 8 / < ( 8 * 8 /

A

B

C

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Vector Calculus

A field can be defined as a function that specifies a value for a particular quantity everywhere in a region. It could be a scalar, vector or other type of field.

– Air temperature in a room: every location has a specific temperature so temperature can be considered as scalar field.

– Wind speed: Speed of air in the atmosphere is another example, since wind speed has a direction (velocity), it is a vector field.

16Wind speed at sea level during hurricane Katrina

Fields:

Gradient of a scalar Field

• Suppose a certain scalar field (i.e. temperature) given by T(x,y,z)

temperature at the point P1 is T(r) = T(x,y,z)

temperature at the point P2 is T(r+dr)=T(x+dx,y+dy,z+dz)

The displacement from P1 to P2 is the displacement vector dr with

components (dx,dy,dz). �� = dx�� + dy�� + dz

Temperature difference between P1,P2: dT = T(r+dr)-T(r)

dT=T(x+dx,y+dy,z+dz) - T(x,y,z)

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X Y

Z

P1(r)

P2(r+dr)

dx

dy

dz

So the change in temperature dT is given as the projection of a ‘change of temperature vector’ (inside square brackets) corresponds to the displacement dr .

This vector is called the Gradient of the scalar T,

written as Grad T or >?.

It is a generalization of one the dimensional differential operator

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to 3 (or higher) dimensions:

Gradient operator

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• Gradient is the ‘slope’ of a scalar field at a point. It gives the direction and magnitude of the greatest rate of change of the field.

• Rate of change in any other direction is given by the projection of gradient in that direction

• Rate of change in a given direction = �� · >?,

�� unit vector in that direction

>@

A surface of constant @ (a level surface), >@is normal to it.

>?, maximum rate of change in this direction

T(x,y)

T lowT high

rate of change in this

direction = �� · >?

��

9�

T constant curves

P

Example:

• Find the directional derivative of f(x,y,z)= x2 +y2 +z2 along the direction 3x � 2y ; zand evaluate it at the point (2,1, 2).

Lets denote the given direction as a

The unit vector in the direction of a= F =F

F=

GHIJ�KL

GMIMI�KN�M=

GH�IJ�KL�

NO

directional derivative in the direction 3x � 2y ; z = P · >Q

=GHIJ�KL

NO· 2xx � 2yy � 2zz =

RHIO�KL

NO

at the point (2,1,2) = R⋅IO⋅NK⋅

NO=

N

NO

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‘Del’ Operator

can be considered as the operator

acting on the scalar field T.

It is called the ‘del’ operator when grad T is written as >? it represent this process.

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Divergence of a Vector Field• The divergence of a vector field at a given point is the net

outward flux per unit volume. It is a scalar quantity.

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flow of A integrated over the surface

Net outward flow,positive divergence

No net outward flowzero divergence

Net inward flow,positive divergence

It represents the amount of field sources at each point.

• Suppose a vector filed A =(Ax,Ay,Az) at the the point (x,y,z).

Lets calculate the net outward flus of this field over a small rectangular box of size (2∆x,2∆y,∆z) centered at (x,y,z)

On the left face of the box outward flux is

Divergence

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X

Z

areafield normal to face at the center of face

P(x,y,z)

2∆x

2∆y

2∆z(x+∆x,y+∆y,z+∆z)

A

(x-∆x,y-∆y,z-∆z)

Az

AyAx 2∆x

2∆z�x,yx-∆y,z)

On the right face of the box outward flux

Y

Total outward flux from left and right faces

Similarly

Total outward flux from front and back faces

Total outward flux from top and bottom faces

Total outward flux

Divergence

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P(x,y,z)

2∆x

2∆y

2∆z

A

X

Z

Y

Divergence of the vector field A

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Divergence Theorem:

The total outward flux of a vector field A at the closed surface S is

the same as volume integral of divergence of A.

Here is the surface integral of A over the surface

flux through the surface element of area ∆S

∆W: directed area element (direction of area is the normal direction)

A

:�∆S

"

The Curl of a Vector Field

Since the del operator > = X

Y

Y�+Z

Y

Y�+

Y

Y�is a vector, it is possible to

take the vector product of it with a vector field.

The vector field produced by this operation is called the curl of the vector field.

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The Curl of a Vector Field

• The curl of a vector field describes the infinitesimal rotation (circulation) of the vector field.

• A good measure of the circulation is the line integral of the field around a closed curve at a given point. If the vector field is ‘rotational’ it contributes to the integral.

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"A

is the line integral of the field A along the closed

curve C, ∮ ( · ∆\ ]

is the integral of component

(projection) of the vector field along the curve.

For a small segment of the curve ∆l :

∆\ ( cos " = ( · ∆\

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no rotationrotation exits away from the center

bulk rotation around center, no local rotation elsewhere

local rotation everywhere

Watch the video: www.youtube.com/watch?v=vvzTEbp9lrc

To see this lets calculate the circulation of field A around a closed rectangular contour abcd of size 2∆Y82∆Z around point P(x,y,z), perpendicular to X axis.

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X

Z

Y

P(x0,y0,z0)

b

(x0,y0+∆Y,z0-∆Z)

c (x0,y0+∆Z,z0+∆Z)d

(x0,y0-∆Z,z0+∆Z)

a

(x0,y0-∆Y,z0-∆Z)

2∆Y

2∆Z

evaluated at (x0,y0,z0)

The Curl of a Vector Field

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curl of the vector field A can be defined as:

C

∆l ∆S is the area enclosed by an enclosed curve C oriented such that the integral has maximum value.

:� : unit vector normal to area ∆S, in the direction of motion of a right handed screw when it is turned in the direction of integral is taken.

"A

∆S

:�

area of loop

Stokes’ Theorem• Stokes’ theorem converts the surface integral of the curl of a

vector field over an open surface into a line integral of the vector field along the curve bounding the surface.

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dS

C

dlValidity of this can be intuitively seen directly from the definition of curl

Divide the surface in to a large number of small areas and apply above to each and take the sum

• the line integrals along the common sides

of adjacent areas mutually cancel.

• only those sides in the periphery of the

surface contribute to the sum.

∴ In the limit areas are infinitesimal this becomes the Stokes’ theorem

Laplacian Operator

The Laplacian operator is the scalar product of the del operator with itself.

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The result is a scalar operator. It can be applied to a scalar or vector field.

Coordinate Systems

Infinitesimal volume element = dxdydz

Infinitesimal volume element dr=dx��+ dy�� + dz

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X

xy

Y

Z

z

P

��

��

O

Cartesian coordinates :

• Three mutually orthogonal axes X,Y,Z,

unit vectors �� , ��, arein the direction of

increasing coordinate value.

A point P in space is given by the

projections x , y , z on coordinate axes.

P(x , y , z) = x��+ y�� + z

−∞ < � < ∞, −∞ < � < ∞, −∞ < � < ∞

P(x,y,z)

dx

dy

dz dr

Cylindrical Coordinates

• Unit vectors e, f�, are in the direction of increasing coordinate values.

• Unlike in Cartesian system not all unit vectors are fixed. Directions of eand f� are depend on the position (azimuthal angle g).

Relation between Cartesian and cylindrical coordinates.

� = h cosg , y = s sing, � = �

e = ��cosf � ��sinf , f� = ;��sinf � ��cosf

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X

x

y Y

Z

z

P

O

∅

s

s

e

f�

• In cylindrical coordinates position of a point P is given by:

– S : the radial distance from Y axis

– ∅ ∶the azimuthal angle, measured from the X-axis in the XY plane

– z : the distance from the XY plane

(same as in the Cartesian system)

0 d h d ∞, 0 d g d 2k, ;∞ d � d ∞

• Sides of the infinitesimal volume element: ��; �h; h�g

• infinitesimal volume element= h�h���g

• Del operator:

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s+ds

sdφ

ds

s

Spherical Coordinates

• Unit vectors �, f�, m� are in the direction of increasing coordinate values. Their directions depend on the position.

Relation between Cartesian and spherical coordinates.

� = $ sin " cosg , y = $sin " sing , � = $ cos "

�� = ��sin" cosg � �� sin " sing � cos " �� = �sin" cosg � m� cos " cosg ; f� sing

m� = ��cos" cosg � �� cos " sing ; sin " �� = �sin" sing � m� cos " sing �f� cosg

f� = ;��sing � ��cosg = �cos " ; m�sing36

X

x

y

Y

Z

z

P

O

∅s

m�

�

f�

In cylindrical coordinates position of a point P is given by:

– r: radial distance the origin

– ∅ ∶azimuthal angle, measured from the X-axis in the XY plane

– " : angle between the Z axis and the line from origin to point P

0 d $ d ∞, 0 d g d 2k, 0 d " d k

" r

X

x

y

Y

Z

z

P

O

∅s

m�

�

f�

" r

Line element: =dr��rdθm�+rsinθdφf�

Volume element: dr rdθ rsinθdφ = r2sinθdθdφ

Del operator :

Example: A sphere of radius 2 cm contains a volume charge density ρgiven by

Find the total charge Q contained in the sphere 37

X

x

y

Y

Z

z

P

O

∅s

m�

�

f�

" r

dr

rsinθdφ

rdθ

Summary

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Show that: curl of gradient of a scalar field is zero > 8 >@ = 0

Show that : divergence of curl of a vector field is zero > · > 8 n = 0

Calculate grad of 5

o

Calculate the divergence of : �

oM

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the Fundamental Theorems

(1)Gradient :

combine (1) and (3)

combine (3) and (2)

(2)Divergence :

(3)Curl :

curl of grad zero

div of curl zero

when the boundary

shrinks to a point

Few basic identities for grad, div, curl

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> · ( 8 p =n. > 8 p − p · > 8 n :

but it leads to contradiction with the divergence theorem , say applied over a sphere:

Naive calculation gives:

Problem is the field is ∞atr=0andnotcorrectlyexpressed

x > y z�v =|}~o

0

Delta function:

Delta Function

To work with such situations the Dirac delta function is � used:

In 1dimention it is defined as:

It is an even fiction with unit area

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1

1

x �

K�� = 1

Q � = lim⇢1

k K���M

Other properties of delta function

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3D Delta function

Which can be generalized to 3 D as

and

Now consider

Since according to divergence theorem and

> · = 0for$ < 0

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• Since for any sphere arbitrary small, it shows

that the entire contribution comes from the point at the origin r=0

Which implies:

So that

As required by the divergence theorem

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Helmholtz Theorem(The fundamental theorem of vector analysis)

The Helmholtz theorem states that any continuous vector field can be written as a sum of a gradient of a scalar field and a curl of a vector field.

�� = >�$� � > ×(r)U is called the scalar potential and W is called the vector potential of the field Proof:

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rr’

0

dv’

P

(Both U(r), W(r) have to go to zero faster than 1/r2 as r →∞)

• So any vector can be written as a sum of a Divergence less field and a Curl less field.

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The divergence and curl of a vector field uniquely define a vector field.