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= Ax (x,y,z dydz) Ax (x - )x,y,z dydz1 1'

+ Ay (x,y+ ) Ay )y,z dxdz (x,y,z dxdz2 2'

+ Az (x,y,z+ ) Az )z dxdy (x,y,z dxdy3 3'

( ) ( -x,y,z)A x,y,z A x x x x y z + x

( ) A x,y,z( )A x,y,z + z z z + z

Ax Ay Az V x

+y

+z

A dSi A A Adiv A = lim

S

=x

+y

+z

V 0 V x y z

Del Operator: = i

x+ i

y+ i

zx y z

Ax Ay Azdiv A= iA = + +x y z

2. Gauss Integral Theorem

A x,y + y,z A x,y,z y

y ( ) y ( )

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N

A dS = i ii A dSS i=1 dS

N i

N

= lim (iA)ViN V 0 i=1

n

= iA dVV

V iA dV = S A ida

3. Gauss Law in Differential Form0 E ida= i(0E dV =) dVS V V

i 0E = ( ) H ida= i H dV = 0 0 ( 0 )

S V

0H = 0i( )II. Stokes Theorem

1. Curl OperationA ds = Curl A( ) i i da

C S

A ds iCurl A = lim C( )

n n 0 dada n

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A ida = A idsS C

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x+x y+ y x A ids = A (x, y dx +) Ay x + ) A x, y + )( x, y dy + x ( y dxxC x x+ x y

1 2 3

y

+ x, y dy Ay ( )y+ y

4

A x, y( ) - A x, y + ) A x + x, y) - A x, y( ) x x ( y y ( y = x y + y x

Ay Ax =daz - x y A ds

xCurl A =( )z=

da

i A

x

y-

A

yz

By symmetry

( ) = A dsi A

-A

Curl A = x zy da z xy

Curl A = A dsi

=Az -

Ay( )x dax y z

A Ay A A Ay A Curl A = i

x z - + i y x - z + i z - x y z z x x y

i x i y i z

= det x y z Ax Ay Az

= A

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2. Stokes Integral Theorem

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N

lim A dsi = A dsiiN i=1 dCi C

N = (A)idai

i=1

= (A)idaS

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3. Faradays Law in Differential Formi = (E)ida = - d H idaE ds dt 0C S S

HE = -0

t

4. Ampres Law in Differential Form

H ds = H ida = J da+ d 0 E idai i dtC S S SH = J+ 0 Et

III. Applications to Maxwells Equations

1. Vector Identitylim A ids =0 = A ida= i A dV ( ) ( )C 0

C S V

i(A)= 02. Charge Conservation

iH = J + 0 E t

0 = iJ + 0 E t

0 = iJ +

t

3. Magnetic Field H iE=-0 t

0 = -t

0 i(0H)= 0i H

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4. Vector Identity

b

E dl = a i ( ) ( ) ba

if a=b

i = ( ) (a)= 0E dl a C

E = idl = 0C

f d f d = f 0( ) i a= i l 0 ( )=S C

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IV. Summary of Maxwells Equations in Free Space

Integral Form Differential Form

Faradays Law

d H E dli = 0 H da E = 0idt tC S

Amperes Law

H dl J da+ 0 i 0i = i d E da H =J + Edt tC S S

Gauss Law E da ( 0 )i = dV i E = 0S V

0i 0 i(0H)= 0H da =S

Conservation of charge

d C

J da+ dt V

= 0 iJ +t

= 01. i dV

2. S

J + 0

E

t

ida = 0 i

J + 0

E

t

= 0

EQS Limit MQS Limit

H E 0, E = E = 0 t iE = i = = (Poissons Eq.) H =J

0( ) 2

( ) (x ',y ',z ') dx'dy'dz' x,y,z =x',y',z' 40

(x x ')2

+(y y ')2

+(z z ')2

i( H)= 0 H = A 0 02 A = 0 J, iA =0

0 J x ',y ', z' dx dy dzA x,y,z( )= ( ) ' ' '

x',y',z' 4 (x x ')2

+(y y ')2

+(z z ')2

12

12

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