p361 lecture2 vector differential integral operators

8/13/2019 P361 Lecture2 Vector Differential Integral Operators

1/28

P361: Electromagnetic

Theory

Patrick Sibanda

Vectors

Divergence and Stokes Theorem


2/28

Vector analysis


3/28

Multiplication of vectors

!Two different interactions (-- the difference?)"Scalar or dot product :

!the calculation giving the work done by a force during adisplacement!work and hence energy are scalar quantities which arise

from the multiplication of two vectors

!ifA!B= 0 The vectorAis zero The vector Bis zero != 90

A ! B =| A || B | cos! = B ! A

!

A

B


4/28

"Vector or cross product :! is the unit vector along the normal to the plane

containing A and B and its positive direction is

determined as the right-hand screw rule

!the magnitude of the vector product ofAand Bisequal to the area of the parallelogram formed by A andB

!if there is a force Facting at a point Pwith positionvector rrelative to an origin O, the moment of a force F

about Ois defined by :

!ifA xB = 0 The vectorAis zero The vector Bis zero != 0

A! B =| A || B | sin!n

!

A

B

A! B = -B! A

L = r! F

n


5/28

Commutative law :

A ! B = B ! A

A! B = -B! A

Distribution law :

A ! (B+C) = A ! B+ A !C

A! (B+C) = A! B+ A!C

Associative law :

A !BC !D = (A !B)(C !D)

A !BC = (A!B)C

A!

B"

C=

(A!

B)"

CA! (B!C)" (A!B)!C


6/28

Reminder --unit vector relationships

Some important relations between the unit vectors alongthe axial directions in terms of the unit vectors and

x! y = y

!z = z

! x = 0

x! x = y ! y = z! z =1

x! x

= y! y

= z! z

= 0

x! y = z

y! z = x

z! x = y

A=

Ax

x+

Ay

y+

Az

xB =Bx x +By y+Bz z

A !B =AxBx +AyBy +AzBz

A"B =

x y z

Ax Ay Az

Bx B

y B

z

x, y z


7/28

Differentiation of vectors

If a vector ris a function of a scalar variable t, then when t varies by an

increment !t, r will vary by an increment r.

ris a variable associated with rbut it needs nothave either the

same magnitude of direction as r:

lim!t!0

!r

!t=

dr

dt

but r= xx+ yy+ zzthusdr

dt=

dx

dtx+

dy

dty+

dz

dtz

NOTE:

d

dt(A ! B) = A

dB

dt+

dA

dtB

d

dt(A" B

)= A"

dB

dt+

dA

dt" B


8/28

As tvaries, the end point of the position vector rwill trace out a curve in space.

Takingsas a variable measuring length along this curve, the differentiation process

can be performed with respect tosthus:

dr

ds=

dx

dsx+

dy

dsy+

dz

dsz

dr

ds= dx

ds!"# $

%&

2

+ dyds

!"# $

%&

2

+ dzds

!"# $

%&

2

=

(dx)2+ (dy)

2+ (dz)

2

ds

=1

dr

dsis a unit vector in the direction of the tangent to the curve

d2r

ds2

is perpendicular to the tangent .dr

ds

d2r

ds2The direction of is the normal to the curve, and the two vectors defined

as the tangent and normal define what is called the osculating plane of the curve.


9/28

!Temperature is a scalar quantity whichcan depend in general upon three

coordinates defining position and a fourth

independent variable time.

" is a partial derivative." is the temperature gradient in the x direction

and is a vector quantity." is a scalar rate of change.

x

T

!

!

x

T

!

!

t

T

!

!

Partial differentiation of vectors


10/28

!A dependent variable such as temperature,having these properties, is called a scalar point

function and the system of variables is

frequently called a scalar field."examples are concentration and pressure.

!There are other dependent variables which arevectorial in nature, and vary with position. These

are vector point functions and they constitutevector field.

"Examples are velocity, heat flow rate, and mass transferrate.

Scalar field and vector field


11/28

The symbol

-- is read as del operator and is expressed as :

Tis a vector in the direction of the most rapid change of T,

and its magnitude is equal to this rate of change.

! = x"

"x+ y

"

"y+ z

"

"z

--it is of vector form and can be used if we wish to know how a continuousand differentiable function (e.g T(x,y,z)) changes over infinitesimal

distance dl

--The Hamilton operator


12/28

The operator is of vector form, a scalar product can be obtained as :

!"A = x #

#x

+ y#

#y

+ z#

#z

$

%

&'

(

)" (Axx +Ayy+Azz)

=

#Ax#x

+

#Ay#y

+

#Az#z

application

The equation of continuity :

!

!x(!u

x)+

!

!y(!u

y)+

!

!z(!u

z)+

!!

!t= 0

where #is the density and uis the velocity vector.

0)( =!

!+"#

t

u$

$

Output - input : the net rate of mass flow from unit volume

Ais the net flux ofAper unit volume at the point considered, counting

vectors into the volume as negative, and vectors out of the volume as positive.

A ! B =AxBx +AyBy +AzBz


13/28

!GradientDefinition. is a vector point-function that derivesfrom a scalar point-funtion

Physical meaning: is the localvariation of !along dr.

!(x,y,z) = l

grad!

grad! =

!!!x

!!

!y

!!

!z

"

#

$$$$$

$$

%

&

'''''

''

= (!, with ( =

!!x

!

!y

!!z

"

#

$$$$$

$$

%

&

'''''

''dr

grad!dr

Consider !(x,y,z) a differentiable scalar field

!Is a vector whose magnitude and direction are those of maximum spacerate of change of !


14/28

!DivergenceDefinition is a scalar point-function derived from a vectorpoint-function It is the spatial derivative of a vector field

Consider -a differentiable vector field then

x x+dx

divv =!"v = #vx#x

+

#vy

#y+

#vz

#z

v(x,y,z)

v(x,y,z) v(x +dx,y,z)

Define divergence of vector field vat a point

as the net outward flux of vper unit volumeas the volume tends to zero

divv =limdv!0

v "dss

#dv

P(x,y,z)

The value of vxat the center of the right hand face -taken to be the avrge over face Can show that the flux of vector vthru right hand face is And that thru the left hand face is

Adding all the faces up leads to

d!R = vx +"vx"x

dx

2

#

$%

&

'(dydz

d!L =" vx"#vx#x

dx

2

$

%&

'

()dydz

d!tot ="v

x

"x+

"vy

"y+

"vz

"z

#

$%&

'(dxdydz

d!


15/28

Ain Aout

! " A> 0

The flux leaving the one end must exceed the flux entering at the other end.

The tubular element is divergent in the direction of flow.

Therefore, the operator "$is frequently called the divergence :

! " A= div ADivergence of a vector

Net outward flow thru surface bounding the volume indicates presence ofsource


16/28

Divegernce Theorem

The volume integral of the divergence of a vector field equals the totaloutward flux of the of the vector through the surface that bounds thevolume

! " vdvv

# = vs

# "ds


17/28

!CurlConsider

Physical meaning: is related tothe localrotation of the vector field:

a(x,y,z) is a differentiable vector field

curla =!" a = det

x y z

#

#x

#

#y

#

#z

ax

ay

az

$

%

&&&&&

'

(

)))))

= x!a

z

!y+

!ay

!z

"

#$

%

&'+ y

!ax

!z+

!az

!x

"

#$

%

&'+ z

!ay

!x+

!ax

!y

"

#$

%

&'

curl v = 0

curl v ! 0

curl v

Curlais defined as a vector whose magnitude is

the maximum net circulation of aper unit area as

the area tends to zero and whose direction is anormal to the direction of the area


18/28

!" A=

x y z

#

#x

#

#y

#

#z

Ax Ay Az

is the curl of a vector ; !"A= curl A

What is its physical meaning?

Assume a two-dimensional fluid element

ux

uy

!x

!y uy

+

!uy

!x!x

ux+

!ux

!y!y

O A

B

Regarded as the angular velocity of OA, direction :

Thus, the angular velocity of OA is similarily, the angular velocity of OB isz!u

y

!x!

z

"ux

"y

!"u =

x y z

#

#x

#

#y0

ux

uy

0

= z#u

y

#x$#u

x

#y

%

&'

(

)*

z

Can be derived starting from the line integral(e.g for an infinitesimal path in the xy-plane)

A !dl" = Axdx" + Aydy"

=

#Ay

#x$#Ax

#y

%

&'

(

)*dxdy

True if the lineintegral runs in the

+ dir of xy-plane


19/28


20/28

A !dl"

=

#Ay

#x$#Ax

#y

%

&'

(

)*dxdy

Stokes Theorem

We showed earlier thatIf we chose a vector dSpointing in the direction of advance of a right-handscrew turned in the direction chosen for the line integral then we can write

A !dl" =#Ay

#x$#Ax

#y

%

&'

(

)*dxdy = (+,A) !dS True only if path is so small that is

nearly constant

!" A

Otherwise we divide the surface into elements of area for which this holds The sum of the many areas is then the integral of the over the finite

surface Thus

Where Sis the area of any open surface bounded by the curve C This is Stokes Theorem

(!"A) #dS

A !dlC

" = (# $ A) ! dSS

"


21/28

!Laplacian: definitions

1 Scalar Laplacian. !(x,y,z) is a differentiable scalar field

2 Vector Laplacian. v(x,y,z) is a differentiable vector field2 2 2

2 x x xx 2 2 2

2 2 2y y y2

y 2 2 2

2 2 22 z z z

z 2 2 2

v v vv

x y z

v v vv

x y z

v v vv

x y z

! " " "#$ = + +# " " "

# " " "##$ = + +%

" " "##

" " "#$ = + +#" " "#&

!v = !vxx+!v

yy+!v

zz

!!="

2

!= div(grad!) =#

2!

#x2 +

#2!

#y2 +

#2!

#z2


22/28

!Laplacian: physical meaningAs a second derivative, the one-dimensionalLaplacianoperator is related to minima andmaxima: when the second derivative ispositive (negative), the curvature is

concave (convexe).

In most of situations, the 2-dimensional

Laplacianoperator is also related to localminima and maxima. If vEis positive:

E

E

v : maximum in E ( (E) > average value in the surrounding)

v : minimum in E ( (E) < average value in the surrounding)

#

x

&(x)

concave

convex

'(


23/28

Coordinates other than cartesian

!Cylindrical(r, !, z) & Sphericalcoordinates

! the edge of the increment element is general curved."If are unit vectors defined as point P:

!r = !rr+ r!""+!z z

dr ! " = 0!= r!dr

! ="

"r

r+1

r

"

"!

!+"

"z

z

r,!,z

(r,!,")


24/28

The gradient of a scalar point function U : in Cylindrical --

!U="U

"r

r+1

r

"U

"!

!+"U

"z

z

! " A=1

r

!

!r(rA

r)+

1

r

!

!!(A

!)+

!

!zA

z

Assuming that the vector Acan be resolved into components in terms of

A = Arr + A

!

!+ Az z

!"A=

1

r

#Az

#! $

#A!

#z

%

&'

(

)*r +

#Ar

#z $

#Az

#r

%

&'

(

)*!+

1

r

#(rA!)

#r $

#Ar

#!

%

&'

(

)*z

2

2

2

2

2

2 11

z

UU

rr

Ur

rrU

!

!+

!

!+"

#

$%&

'

!

!

!

!=(

)

r,!and z


25/28

The gradient of a scalar point function U : in Spherical

!U="U

"r

r +1

r

"U

"!

!+1

r sin!

"U

""

z

! " A=1

r2

!

!r(r

2A

r)+

1

rsin!

!

!!(A!sin!)+

1

rsin!

!

!"A"

Assuming that the vector Acan be resolved into components in terms of :

A = Arr + A!

!+ A""

!"A=

1

rsin!

#

#!(A

!sin")$

#A!#"

%

&'

(

)*r+

1

rsin!

#Ar

#! $ sin"

#

#r (rA

!)

%

&'

(

)*"+

1

r

#

#r(rA!)$

#Ar

#!

%

&'

(

)*"

2

2

222

2

2

2

sin

1sin

sin

11

!"""

"" #

#+$

%

&'(

)

#

#

#

#+$

%

&'(

)

#

#

#

#=*

U

r

U

rr

Ur

rrU

r,!and "


26/28

3. Differential operators

!SummaryOperator grad div curl Laplacian

is a vector a scalar a vectora scalar

(&a vector)

concerns

a scalar

field

a vector

field

a vector

field

a scalar field

(& a vector field)

Definition !! ! "v !" v !2!& !

2v


27/28

Gauss Divergence Theorem

Stokes TheoremA !dl

C" = (# $A) ! dSS"

A ! dSS

" = # ! Advv

"


28/28

Useful equations about Hamiltons operator ...

)()( ABBAABBA

BAABBA

!"!+!"!+"#+"#=

"#+"#=#"

Ais to be differentiated

UUU !"+"!="! AAA

UUU !"#"!="! AAA

BAABBA !"#$!"#=!#"

ABBABAABB)A !"#"!#!"+"!=$$" (

BABABA !"#!"=$!$ )(

ABABAB !"#!"=$!$ )(

BA =)(

2

1AAAAA

2!"!+"#="

0U

0

=!"!

="!#!=#!"!

!$#!!="!"!

AAAAA(

2

)valid when the order of differentiation is not

important in the second mixed derivative

p361 lecture2 vector differential integral operators

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