A REVIEW ON VECTOR ALGEBRA & CALCULUSeee.guc.edu.eg/Courses/Communications/COMM402...

ELECTROMAGNETIC PROF. A.M.ALLAM

A REVIEW ON

VECTOR ALGEBRA &

CALCULUS

EMF


Vector quantity: has both magnitude and direction as velocity, force, electric

field

1-Vectors and scalars

A

A

1

azzyyxx aAaAaAaAaAA ˆˆˆˆ ˆ||

222|| zyx AAAAA

||ˆ

A

Aa

&

Scalar quantity: has magnitude only as temperature, mass, volume or energy

aAA ˆ


Position vector:

zyx a za ya xr

zyx a za ya xr

Let the position of source (S) is defined by:

and the position of observation point P is

defined by:

rrR

Thus, the distance (R) between source and observation point is:

222 )()()( zzyyxx

R

Ra R

x

z

y

S

r

r

rrR

)z ,y ,x(S

)z ,y ,x(P

Ra Rzyx a )(a )(a )( zzyyxx

|||| rrRR


2-Vector algebra

Vector addition and subtraction:

The following two vectors:

zzyyxx a Aa Aa A A

BA

zzyyxx a Ba Ba B B

&

- Addition:

zzzyyyxxx a B Aa B AaB A BA

A

B

- Subtraction:

zzzyyyxxx a B Aa B Aa B A BA BA

B

A


Vector multiplication:

- Scalar (dot) product:

c o s || || . BABA

The scalar (dot) product between two vectors is defined as the product

of the magnitudes of , and the cosine of angle between them. The result

is scalar

A

BA

&

B

For

zzyyxx a Aa Aa A A

zzyyxx a Ba Ba B B

&

zzyyxx B AB AB A B . A

A

B

c o sA


- Vector (cross) product:

n s in || || BABA

zyx

zyx

zyx

B B B

A A A

a a a

B A

The vector (cross) product between two vectors is defined as the

product of the magnitudes of , and the sine of angle between them. The

result is vector (which is perpendicular to the plane of )

A

B&A

B

B&A

A

Bn

ya

xa

za

+

+ +

_ _

_

Twice the area of triangle between

the two vectors.

B

BA

A


zyx

zyx

zyx

C C C

B B B

A A A

)CB( . A

A. Triple scalar product:

, a Aa Aa A A zzyyxx

zzyyxx a Ba Ba B B

zzyyxx a Ca Ca C C a nd

) B A( . C) A C( . B) C B( . A

B. Triple vector product:

) C B( AC )B A(

)C . B(A)C . A(BC )B A(

Triple product of vectors:

)B . A(C-)C . A(B) C B( A

A

C

n

B

A · (B × C) = ABC sinαcosβ

The volume of the

parallelepiped

A × (B × C) = B(A · C) − C(A ·B)

This expression is known as the

“BACK-CAB”-rule.


3-Coordinate systems:

• An orthogonal system is one in which the coordinates are mutually perpendicular. The most standard and commonly used are:

– Cartesian (or rectangular)

– Circular (cylindrical)

– Spherical

The solution of a given practical problem can be greatly facilitated by the proper choice of a coordinate system that best

fits the geometry of the problem


x

z

y

1- Cartesian coordinates (x, y, z):


xx a d yd zSd

yy a dxdzSd

zz a d xd ySd

Position vector:

zyx a a a zyx AAAA

Differential length:

zzyyxx a da da d d

Differential area:

xa dydzSd x

ya dzdxSd y

za d y d xSd z

Differential volume: d x d y d zd


x

z

y

φρ

2- Cylindrical coordinates ( , , z):


Position vector:

za a a zAAAA


za a a d zddd

Differential area:

a d zdSd

a dzdSd

za ddSd z

Differential volume: d zddd

a dzd Sd

a dzdSd zz a dd Sd

d

zddd d

d

d d

dz

dz


x

z

y

φ

θ

r

3. Spherical coordinates (r , , ):

20

0

0

r


Position vector:

a a a AAAA rr


a s ina a drrdd rd r

Differential area:

rr ddrSd a s in 2

a s in d rdrSd

a d rdrSd

Differential volume:

dd rdrd s in 2

d s in r

ddrd sinrd 2

d r

rdr


Transformation between coordinate system

xa

ya a

a

x

y

c o sx

s in y

zz

22 yx

x

ytan 1-

cos sin 0

-sin cos 0

0 0 1

Cart.

Cyl. xa yaza

a

a

za

Example:

a s ina c o s a x

yx a s ina c o s a

1- Cartesian Cylindrical :

zz


2-Spherical Cylindrical :

a

za ra

a

z

r

inr s

o srz c

22 zr

ztan 1-

sin cos 0

0 0 1

cos -sin 0

Sph.

Cyl. ra a a

a

a

za

Example:

za c o sa s in a r

a c o sa s in a r


3- Cartesian Spherical

za c o sa s in a r

sin cos cos cos -Sin

sin sin cos sin Cos

cos -sin 0

Sph.

Cart. ra a a

xa

ya

za

zx sa a c o s)a inˆ (c o s s in y

zx sao s a c o sa ins inˆ c s in y

22222 zyxzr

ztan

ztan

22

1-1- yx

x

ytan 1-

o sinr c s x

inry s s in

c o s rz

za s ina c o s a

zx sa a s in )a inˆ (c o s c o s y

zx sao s a ina s in c o sˆ c c o s y

yx a c o sa s in a

Example:


• Line integral : (integration on path)

zzyyxx dAdAdAd .A

A

Let zzyyxx aAaAaA ˆˆˆA

zzyyxx adadad ˆˆˆd

c

d .A

•Surface integral :

ss

dSnAS ˆ .d .A

s

Sd .A S

A

n(Flux of vector field )A

(Total outward Flux)

•Volume integral : v

dv A

c

S

v

Circulation (rotation) of around CA

d

Path

P1

P2

4-Vector calculus


• Differential operator (Del operator):

zyx az

ay

ax

ˆˆˆ

It does not have any physical meaning, but once applied to a scalar physical quantity it attains a

meaning, be definition:

Or it is vector whose magnitude is equal to the maximum rate of change of the physical

quantity per unit distance and whose direction is along the direction of maximum increase

Is the maximum space rate of change of physical function and its direction is being that in

which the change is most rapidly or maximum

•Gradient

Source

ψ1 <Ψ2 <Ψ3

The at a point is ┴ the

constant V surface that passes

through that point

V

The projection of in the

direction of a unit vector is

called the directional derivative

of V along

Vk

k

kV ˆ.


•Divergence of a vector

• The divergence can operate only on vectors and the result is a scalar

• Positive divergence means volume V contains the source of the flux

• Negative divergence means volume V contains the sink of the flux

•Uniform field is divergenceless (Solenoid field)

The divergence of the vector field is the ability of this field to diverge from or

converge to a point in space

V

Sz

A

y

A

x

A

V

Sd.AA divA . zyxS

0Vlim

z

A

y

A

x

A

V

Sd.AA divA . zyxS

0Vlim


Div A is zero if the same amount of flux enters into the

imaginary infinitesimal volume centered at that point

as leaves

At a given point in space:

Vector A has a positive divergence, if the net flux flowing

outward through the surface of an imaginary infinitesimal

volume centered at that point is positive

Vector A has a negative divergence, if the net flux into

the imaginary infinitesimal volume centered at that

point is negative


Properties of divergenceFormula of divergence

B . A .)B A( .

. A A .

)A .( A .)() A( .

It states that the total outward flux of a vector field A through the closed

surface S is the same as the volume integral of the divergence of A

Divergence theorem (Gauss theorem)

Sd . Adv A .sv

V

S

A


•Curl of a vector

The curl of a vector field A at a point P may be regarded as a measure of the circulation or how much the field curls around P

The curl of physical function is the ability of this function

to circulate or rotate

A A A

z y x

a a a

S

d.AlimmaxA curlA

zyx

zyx

c

0S

The curl of vector A is an axial (or rotational) vector:

-whose magnitude is the maximum

circulation of A per unit area as

that area tends to zero

-and whose direction is the normal

direction of the area when the area

is oriented so as to make the

circulation maximum

The direction out of the

slide


The direction of the curl is n, the unit vector normal of s, defined according to

the right-hand rule: with the four fingers of the right hand following the

contour direction dl, the thumb points along n.

ˆ

ˆ

The curl of uniform field is zero

Formula of curl

-x


Stokes theorem

d . ASd . A cs

The surface integral of the curl of a vector field taken over surface S is

equal to the line integral of the vector field taken over the closed contour C

A

S

A

C

Other differential expressions

0)(

0)A( .

2)( .

A)A.()A( 2

2

2

2

2

2

22

zyx

Laplace’s operator

(Laplacian)


Curl f in arrows Curl f in arrows


Summary on a vector field

0 )A. (a nd 0A

0 )A. (a nd 0A

0 )A. (a nd 0A

0 )A. (a nd 0A

Uniform field

General field

Solenoidal (rotational) field

Irrotational (conservative or potential) field

Generally: any vector field is completely determined by its and A.

A

A REVIEW ON VECTOR ALGEBRA & CALCULUSeee.guc.edu.eg/Courses/Communications/COMM402...

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