Chapter2. Vector analysis

Chapter2 Vector analysis

Chapter2. Vector analysis

2 1 Introduction 2-1 Introduction

1. vector algebra

addition, subtraction and multiplication , p

2. orthogonal coordinate systems 3. vector calculation

differentiation, integration (line, surface and volume differentiation, integration (line, surface and volume integrals)

del( ) operator, gradient, divergence and curl operation ∇

2 2 Vector Addition and Subtraction 2-2 Vector Addition and Subtraction

1 Addition 1. Addition

, , A

A Aa A A a

= = = A

Commutative law : A B C

A B B A + =

+ = + 1

Associative law : 2 A + ( B C + ) = ( A + B ) + C 2. Subtraction

( )

A B − = + − A B

2 3 Products of Vectors 2-3 Products of Vectors

1. Scalar or dot product 1. Scalar or dot product

A B i = AB cos θ

A θ B

Commutative law :

Distributive law : ( ) A B B A

A B C A B A C

=

+ = +

i i

i i i

1 2

θ B

2

or

A A i = A A = A A i

( ) ( )

C = C C i = A + B i A + B

2 2

2 2

2 cos

2 cos

A B AB

A B AB

θ α

= + +

= + −

B C

α

A

2. Vector or cross product 2. Vector or cross product A B × = a AB

sin θ

B

A B ×

A B × = − × B A

θ A

an

ω

V

d

sin V = ω d = ω R θ

θ

d • p

R

V = × ω R

3 Product of three vectors 3. Product of three vectors

( ) ( ) ( )

A B C i × = B C i × A = C A B i ×

( ) ( ) ( )

A × B C × = B A C i − C A B i

2 4 Orthogonal Coordinate Systems 2.4 Orthogonal Coordinate Systems

1 2 3

Base vector ( , , ) u

u

u

1 2 3 2 3 1 3 1 2

1 2 2 3 3 1

( )

, ,

0

u u u u u u u u u

u u u u u u

× = × = × =

= = =

i i i

1 2 2 3 3 1

1 1 2 2 3 3

u u i = u u i = u u i = 1

1 1 2 2 3 3 1 1 2 2 3 3

,

( 1, 2, 3) : const does not mean that the surface in plane. It can

u u u u u u

i

A A u A u A u B B u B u B u u i

= + + = + +

=

be curved surfaces

e.g) spherical coordinate system( , r θ , )

h f f h

φ : const means the surface of a sphere

(it is not plane, but , , are perpendicular to each other) r

r θ φ

A B A B + A B + A B

1 1 2 2 3 3

1 2 3

u u u u u u

A B A B A B A B

u u u

A B A A A

= + +

i

1 2 3

1 2 3

( ) ( ) ( )

u u u

u u u

A B A A A

B B B

A B A B A B A B A B A B

× =

2 3 3 2 1 3 3 1 1 2 2 1

1 2 3

= u A B (

− A B

) − u A B (

− A B

) + u A B (

− A B

)

1. Differential change in one of the coordinates 와 Diffetential length change 의 상관 관계

1 2 3

, ( : metric coefficient, may be a function of , , ) e.g) polar coordinate(the subset of c

i i i i

dl =h du h u u u

ylindrical coordinate) (u u_{1 2})=(r φ) dφ inφ → dl₂ = rdφ inφ direction ( , )=( , ) in in direction

2. differential length change : vector

u u r d dl rd

l

φ φ φ → = φ φ

임의의 방향의

1 1 2 2 3 3 1( 1 1) 2( 2 2) 3( 3 3)

3

dl = dll =u dl +u dl +u dl =u h du +u h du +u h du

Differential volume change : scalar 3

1 2 3 1 2 3

. Differential volume change : scalar dv =h h h du du du

1 1

4. Differential area change : vector (right hand rule)

) Differential area normal to the unit vector ds dsn

cf ds u

=

1 1

) Differential area normal to the unit vector

cf ds u

1 2 3 2 3 2 3

ds =dl dl = h h du du

2 4 1 Cartesian Coordinates 2-4.1 Cartesian Coordinates

di l h h h 1

1 2 3

( , u u u , ) = ( , , ) x y z distance element. so h

= h

= h

= 1

u₁ = x u₂ = y u₃ = z , ,

position vector to the point ( , , )

u x u y u z

x y z

x y z

= = =

× =

1 1 1

_{x x y y z z}

OP x x y y z z A B A B A B A B

= + +

= + +

i

1 2 3

( ,u u u, )=( , , )x y z

_{x y z}

x y z

x y z

A B A A A

B B B

× =

, ,

x y z

x y z

dl xdx ydy zdz

ds dydz ds dxdz ds d

= + +

= = = xdy

dv =dxdydz

2 4 2 Cylindrical Coordinates 2-4.2 Cylindrical Coordinates

( u u u

) = ( r φ z ) h = h = 1 h = r

( , u u u , ) = ( , , ) r φ z h

= h

= 1, h

= r

, , r × = φ z φ × = z r z r × = φ

r z

A = rA + φ A

+ zA

1 2 3

( ,u u u, )=( , , )r φ z

dl d φ φ d d

dl=rdr+φ φd +zdz

, ,

r z

dl rdr rd zdz

ds rd dz ds drdz ds rdrd dv rdrd dz

φ

φ φ

φ φ

φ

= + +

= = =

=

, ,

r z

ds rd sz ds drdz ds rdrd dv rdrd dz

φ φ φ

φ

= = =

=

) _{r z x y z}

cf A=rA +φA_φ +zA ⇒ xA + yA + zA

φ

cos sin

sin cos

y

x r r

y r r

A A x A r x A x A A

A A y A r y A y A A

φ

φ φ

φ φ

φ φ φ

φ φ φ

= = + = −

= = + = +

i i i

i i i

cos sin 0 sin cos 0

0 0 1

x r

y

A A

A A

A A

φ

φ φ

φ φ

⎡ ⎤ ⎡ − ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⇒ ⎢ ⎥ ⎢= ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣A_z 0 0 1⎦ ⎣ ⎦A_z ^대입

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

cos , sin , x=r φ y =r φ z = z

대입

2 2 1

, tan ( ), y

r x y z z

φ ⁻ x

= + = =

2 4 3 Spherical Coordinate 2-4.3 Spherical Coordinate

( u u u

) = ( R θ φ ) ^{h = 1 h = R h = R sin θ} ( , u u u , ) = ( , , ) R θ φ h

= 1, , sin h

= R h

= R θ

, , R × = θ φ θ φ × = R φ × = R θ

A = RA

+ θ A

+ φ A

sindl = RdR+θ θ φRd + R θ θd

2

2

sin , sin ,

sin

dsR R d d ds R dRd ds RdRd

dv R dRd d

θ φ

θ θ φ θ φ θ

θ θ φ

= = =

=

) sin cos sin sin cos

cf x = R θ φ y = R θ φ z = R θ

2 2

2 2 2 1 1

) sin cos , sin sin , cos

, tan , tan

cf x R y R z R

x y y

R x y z

z x

θ φ θ φ θ

θ ⁻ φ ⁻

= = =

= + + = + =

H.W1 : Find the conversion matrix from spherical into cartesian

2 5 Integrals Containing Vector Functions 2-5 Integrals Containing Vector Functions

Fdv vdl F dl A ds

∫

∫

∫

∫

1. (for cartesian)

2 ( )[ ] ( ) ( ) ( )

v c c s

v z y x

Fdv vdl F dl A ds

Fdv Fdxdydz

vdl v x y z xdx ydy zdz x v x y z dx y v x y z dy z v x y z dz

=

= + + = + +

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫

1

2. ( , , )[ ] ( , , ) ( , , ) ( , , )

) from point P

cvdl cv x y z xdx ydy zdz x v x y z dxc y v x y z dyc z v x y z dzc

cf

= + + = + +

∫ ∫ ∫ ∫ ∫

2

to point P2

around path

p vdl C ⇒ vdl

∫

∫

, around path

p vdl C ⇒ cvdl

∫ ∫

y

) evaluate

ex ∫ r dr

P2

(1,1) P

2 2 2

where from the original to the point (1,1) ) along the direct path

r x y P

a OP

= +

1

2

) along the path ) along the path

b OP P

c OP P

x

P1

o

2 2

2 2 2

0 0

-1

) 2 2

3

where cos sin and . since tan tan

a

r dr r r dr r

y y

r x φ y φ φ π φ φ π

= =

= + = = ⇒ = =

∫ ∫

where cos sin and . since tan tan

4 4

2 2

3 3

r x y

x x

x y

φ + φ φ φ ⇒ φ

= +

1

1

1

2 2 2 2 2

0 0

) along O

4 1

( ) (0 ) ( 1)

3 3

P P P

P

b P P

x + y dr = + y dy y + x + dxx = x + y

∫

∫

∫

2 2

0

3 3

1 4

) same way, ( )

3

c ∫

x + y dr = x + ₃ ^y

3 ∫ F dl ⁱ = ?

3. ?

) Given 2 , evaluate

C

B A

F dl

ex F xxy y x F dl

=

= −

∫

∫

i

i ) In cartesian coordinates

) In cylindrical coordinates a

b

) I t i

x=0,y=3

2 2

) In cartesian

( 2 ) ( ) 2

( 2 ) 9(1 ) using 9

a

F dl xxy y x xdx ydy xydx xdy

xydx xdy π x y

= − + = −

− = − + + =

∫

i i

3, 0

( 2 ) 9(1 ) using 9

2 ) In cylindrical

( cos 2 sin ) ( sin 2 cos )

x y xydx xdy x y

b

F r xy φ xy φ φ xy φ x φ

= = = + + =

= − − +

∫

where x =rcos , φ y =rsin , φ dl =φr 3

3( sin 2 cos )

d d

F dl xy x d

π

φ φ φ

φ φ φ

π

=

= − +

i

2

0 3( sin 2 cos ) 9(1 ), where 3cos and 3sin

xy φ x φ φd π2 x φ y φ

− + = − + = =

∫

4 ∫ A d ∫ A d h i l h fl f h

4. the integral measures the flux of the vector field flowing through the area

s

A ds

A nds

A S

= ⇒

∫ ⁱ ∫ ⁱ

cf ) the convention for the positive direction of or a closed surface

ds n

an open surface

depend on the direction in which the perimeter of the open

f i d

⇒

surface is traversed

disk

1 2

) Given k k , ( 3, 2)

ex ) Given F = a + a , ( z = ± 3, r = 2) evaluate .

r z

s

ex F a a z r

r r

F ds

+ ±

∫ ⁱ

∫ ∫ ∫ ∫

)

a)top face : , , 3 ( 0 2, 0 2 )

top bottom side

S face face wall

sol F ds F nds F nds F nds

n z ds rdrdφ z r φ π

= + +

= = = = → = →

∫

∫

∫

∫

2 2

2 2

0 0

3 12

b)bottom face : , , 3

top face

F nds k rdrd k

n z ds rdrd z

π φ π

φ

= =

= − = = −

∫

∫ ∫

) , ,

φ

2 2

2 2

0 0 3 12

c)side wall : 2

bottom face

F nds k rdrd k

n r ds rdrd r

π φ π

φ

= =

= = =

∫

∫ ∫

2 3

1 1

0 3

c)side wall : , , 2

_side 12

wall

n r ds rdrd r F nds ^π k rdrd k

φ

φ π

−

= = =

= =

∫ ∫ ∫

∫

i

1 2

12 ( 2 )

S F ds π k k

∴

∫

2 6 Gradient of a Scalar Field 2-6 Gradient of a Scalar Field

scalar or vector fields : function of ( t ; u u u

) scalar or vector fields : function of ( ; , , ) space rate of change of a scalar fields at a given time

the gradient of the scalar : the vector that represents both the magnitude and t u u u

g p g

the direction of the maximum space rate of increase of a scalar

grad dV or dV (where, )

V n V n dl dn

dn ∇ dn ≠

) cos ( )

( )

dn dn

dV dV dn dV dV

cf n l V l

dl dn dl dn dn dV V dl

α

= = = = ∇

∇

i i

∵ dV = ∇ i ( V dl )

V V V dV ∂ dl₁+ ∂ dl₂ + ∂ dl₃

1 2 3

1 1 2 2 3 3 1 1 1 2 2 2 3 3 3

where, ( ) ( ) ( )

dV dl dl dl

l l l

dl u dl u dl u dl u h du u h du u h du

V V V V V V

= + +

∂ ∂ ∂

= + + = + +

⎛ ¹ ∂ ² ∂ ³ ∂ ⎞

(

)

1 2 3 1 2 3

1

V V V V V V

dV u u u u dl u dl u dl u u u dl

l l l l l l

V u V

⎛ ∂ ∂ ∂ ⎞ ⎛ ∂ ∂ ∂ ⎞

=⎜⎝ ∂ + ∂ + ∂ ⎟⎠ + + =⎜⎝ ∂ + ∂ + ∂ ⎟⎠

∇ = ∂

∴

i i

2 3 1 2 3

V V V V V

u ∂ u ∂ u ∂ u ∂ u ∂

+ + = + +

V u1

∇ l

∴ ∂₁ ² ₂ ³ ₃ ¹ _{1 1} ² _{2 2} ³ _{3 3}

1 2 3

1 1 2 2 3 3

u u u u u

l l h u h u h u

u u u V

h u h u h u

+ + + +

∂ ∂ ∂ ∂ ∂

⎛ ∂ ∂ ∂ ⎞

=⎜⎝ ¹∂ ¹ + ²∂ ² + ³∂ ³ ⎟⎠

1 2 3

1 1 2 2 3 3

u u u

h u h u h u

⎝ ⎠

∂ ∂ ∂

⇒ ∇ ≡ + +

∂ ∂ ∂

) divergence operation curl opetation

cf ∇ ⇒

∇× ⇒ i

Vector differential operator

∇× ⇒ curl opetation

2 7 Divergence of a Vector Field 2-7 Divergence of a Vector Field

Divergence of a vector field at a point ( )

Net outward flux of per unit volume as the volume about the

A div A

⇒ A

point tends to zero

∫ A d lim

v

A ds div A

Δ →

v

= ∫ Δ ⁱ

→ (two d imensional surface/three dimensional volume) spatial

derivatives as the volume approaches zero pp

. ) In cartesian coordinates

s

e g

A ds

⎡ ⎤

∫ ⁱ

face face face face face face

⎡ ⎤ A ds

= ⎢ + + + + + ⎥

⎣ ∫ ∫ ∫ ∫ ∫ ∫ ⎦ ⁱ

0 0 0

front face

( ) ( , , )

front front front x 2

front face

A ds = A Δs = A x Δ Δ =y z A x + Δx y z Δ Δy z

∫

0 , 0 0

0 0 0 0 0 0

( , )

( , , ) ( , , )

2 2

back face

x

x x

x y z

A

x x

A x y z A x y z high order term

x

∂

Δ Δ

∴ + = + +

∂

_bac A dsi ( ) ( ₀ , ₀, ₀)

2

( ) ( )

back back back x

k face

x

A s A x y z A x x y z y z

A

x x

A A hi h d t

= Δ = − Δ Δ = − − Δ Δ Δ

Δ Δ ∂ +

∫

0, 0 0

0 0 0 0 0 0

( , )

( , , ) ( , , )

2 2

x

x x

x y z

A x y z A x y z high order term

∴ − = − x +

∂

⎛ ⎞

( 0, 0 0, )

[ ] . . .

x y z

x front back

face face

A ds A H O T x y z

x

A A A

⎛ ∂ ⎞

∴ + = ⎜ ⎝ ∂ + ⎟ ⎠ Δ Δ Δ

⎛ ∂ ∂ ∂ ⎞

∫ ∫ ⁱ

( 0, 0 0, )

. . . in , ,

x y z

x y z

s

A A A

A ds x y z H O T x y z

x y z

⎛ ∂ ∂ ∂ ⎞

∴ ∫ ⁱ = ⎜ ⎝ ∂ + ∂ + ∂ ⎟ ⎠ Δ Δ Δ + Δ Δ Δ

2 3 1 1

1 2 3 1 2

1 ( ) (

x

A

A A

div A A h h A h h

x y z h h h u u

∂ ∂ ∂ ∂ ∂

≡ ∇ = + + = +

∂ ∂ ∂ ∂

∴

i ∂

3

) ( )

A h h A

u

⎡ + ∂ ⎤

⎢ ∂ ⎥

⎣ ⎦

depend on the position of the point on which it is evaluated

⎣ ⎦

⇒

∂ ∂ ∂

1 2 3

1 1 2 2 3 3

cf ) u u u

h u h u h u

∂ ∂ ∂

∇ ≡ + +

∂ ∂ ∂

( )

ˆ ˆ ˆ ˆ ˆ ˆ

A ⎛ ∂ ∂ ∂ ⎞ A A A

∇ ⎜ + + ⎟

(

)

( ) ( ) ( )

1 2 3 1 1 2 2 3 3

1 1 2 2 3 3

1 1 1 2 2 2 3 3 3

ˆ ˆ ˆ ˆ ˆ ˆ

A u u u u A u A u A

h u h u h u

u u A u u A u u A

h u h u h u

∇ =⎜⎝ ∂ + ∂ + ∂ ⎟⎠ + +

∂ ∂ ∂

= + +

∂ ∂ ∂

i i

i i i

1 1 2 2 3 3

h u∂ h u∂ h u∂

cos sin , sin cos

r x y x y

r

φ φ φ φ φ

φ

= + = − +

∂ ∂

y

r , φ r φ φ φ

∂ = ∂ =

∂ ∂

) cylindrical coordinate eg

x φ

φ

( ) ( ) ( )

1 2 3 1 2 3

) y

, , and 1, , 1

_{r z}

g

u r u u z h h r h

r rA r A r zA

r r ^φ r

φ

φ

= = = = = =

∂ + ∂ + ∂

∂ i ∂ i ∂ i

x

_{r r} z _{z z}

r r A r r A r A r A r A r z A

r r r ^φ r ^φ r r

φ φ

⎛ ⎞ ⎛ ⎞

∂ ∂ ∂ ∂ ∂ ∂ ∂

⎛ ⎞

= ⎜⎝i∂ ⎟⎠ + i ∂ +⎜⎜⎝ i∂ ⎟⎟⎠ + i ∂ + ⎜⎜⎝i∂ ⎟⎟⎠ + i ∂ = ∂ 1

( )

Ar

r

∂

( )

)1 _{r r} A^r

cf rA A

r r r r

∂ = ∂ +

∂ ∂

2 8 Divergence Theorem 2-8 Divergence Theorem

Adv A ds

∇ =

∫ ⁱ ∫ ⁱ

from definition of divergence, ( A)

j

V s

j j

s

Adv A ds

v A ds

∇

∇ Δ =

∫ ∫

∫

i i

0 0

1 1

lim ( ) lim

j j j

N N

j j s

v v

j j

A v A ds

Δ → = Δ → =

⎡ ⎤ ⎡ ⎤

∇ Δ =

⎢ ⎥ ⎢ ⎥

⎣ ∑ ⁱ ⎦ ⎣ ∑ ∫ ⁱ ⎦

Volume integral

0 1

lim

j j

N

s s

v j

A ds A ds

Δ →

⎡ ⎤

⎢ ⎥ =

⎣ ∑∫ ⁱ ⎦ ∫ ⁱ

( )

V

∇ A dv

∫

ⁱ

_⎣

_⎦

∫

( ∇ A dv ) = A ds

∴ ∫ ∫

^{( i )} ∫ ∫

ⁱ

2 9 Curl of a Vector Field 2-9 Curl of a Vector Field

Flow source outward flux divergence

Vortex source

C

cause a circulation of a vector around it

circulation of around contour C A

∫

circulation will be the work done by the force in moving the object once around the contour

→

electric field e.m.f water whi lr

→

ing down a sink drain → vortex sink

( ) (

)

( )

lim 1

1

n C

curl A A s a A dl

Δ → s ⎡ ⎤

≡ ∇× Δ ⎣

∫

∫

i

( ) ( )

( )

) lim

u u

u s C

u u

cf A a A A dl

Δ → s

∇× = ∇× =

Δ

∫

i i

( )

ˆ ˆ

u , , : , , ,

1

u u

x s x y C

= Δ = Δ Δ

⎛ ⎞

( )

∫

1,2,3,4

lim 1

x y z

A A dl

Δ Δ → y z

⎛ ⎞

∇× = Δ Δ ⎝⎜

∫

0 0 0

side1

, ( , , )

z 2

dl = Δz z A dli = A x y + Δy z Δz

0 0 0

0 0 0 0 0 0

( , , )

where, ( , , ) ( , , ) . . .

2 2

( )

z

z z

x y z

A

y y

A x y z A x y z H O T

y A

A dl A y H O T

∂

Δ Δ

+ = + +

∂

⎧ Δ ∂ ⎫

⎪ ⎪Δ

⎨ ⎬

∫

0 0 0

0 0 0

side1

( , , )

( , , ) . . .

2 side3

z z

x y z

A

A dl A x y z y H O T z

y

∂

⎪ Δ ⎪

∴

∫

A

y ⎧ y ∂ ⎫

Δ ⎪ Δ ⎪

∫

dl _{0 0 0 0 0 0}

side3

( , , )

, ( , , ) ( , , ) . . . ( )

2 2

z

z z

x y z

A

y y

z z A dl A x y z z A dl A x y z H O T z

y

⎧ ∂ ⎫

Δ ⎪ Δ ⎪

= − Δ ⁱ = − Δ ∴

∫

side1 side3+

0 0 0

( , , )

. .

side2 side4

z

x y z

A H O T y z

y

⎛∂ ⎞

=⎜⎝ ∂ + ⎟⎠ Δ Δ

+

. . 0 , when 0 and z 0

H O T → Δ → y Δ →

0 0 0

( , , )

side2 side4

. . ( )

y z y

x x y z

A A A

H O T y z A

y y z

+

∂ ∂

⎛ ⎞ ∂

= −⎜⎝ ∂ + ⎟⎠ Δ Δ ∴ ∇× = ∂ − ∂

General expression in cartesian

z Ay

A x⎛∂A ∂ ⎞ y ∂

∇× = ⎜ − ⎟+ ^{x z y x}

x y z

A A A A

z⎛∂ ∂ ⎞

∂ ∂ ∂ ∂

⎛ − ⎞+ ⎜ − ⎟=

⎜ ⎟

A x y

y z

∇× = ⎜⎝ ∂ − ∂ ⎟⎠+

General expression for orthogonal curvilinear systems

x y z

z x z x y x y z

A A A

− + ⎜ − ⎟=

⎜ ∂ ∂ ⎟ ∂ ∂ ∂ ∂ ∂

⎝ ⎠ ⎝ ⎠

1 1 2 2 3 3

1 2 3 1 2 3

ˆ ˆ ˆ

1

u h u h u h

A h h h u u u

∂ ∂ ∂

∇× = _{1 2 3} ∂ ₁ ∂ ₂ ∂ ₃

1 1 2 2 3 3

h A h A h A

2 10 Stokes’s Theorem 2-10 Stokes s Theorem

( ∇× A ds ) = A dl

∫ ⁱ ∫ ⁱ

from definition of curl, ( ) ( )

j

j j C

A s A dl

∇× ⁱ Δ =

∫

( )

S

∇× A ds =

A dl

∫ ⁱ ∫ ⁱ

0 1

lim ( ) ( ) ( )

j

j

N

j j S

s j

A s A ds

Δ → =

⇒

∑

∫

∑ ∫

∫

0 1

lim

j Cj C

s j

A dl A dl

Δ → =

∑ ∫

∫

) ( ) 0,

cf ∫

∇× A ds ⁱ =

closed surface no boundary contour

S

⇒

∫

2 11 Two Null Identities 2-11 Two Null Identities

1 ∇× ∇ ( V ) ≡ 0

the curl of the gradient of any scalar field is identically zero

1. ( ) 0

[ ] ( ) 0

V

V ds V dl dV

∇× ∇ ≡

→

∇×∇ = ∇ = =

∫ ⁱ ∫ ⁱ ∫

[ ] ( ) 0

cos ( )

S

V ds

V dl

dV

dV dV dn dV dV

n l V l dl dn dl dn α dn

∇×∇ = ∇ = =

= = = = ∇

∫ ⁱ ∫ ⁱ ∫

∵ i i

( )

dl dn dl dn dn

dV = ∇ V d

∵ i l

참 고

the divergence of the curl of any vector field is identically zer

2. ( ) 0

o

∇ ∇× A =

→ i

참 고

the divergence of the curl of any vector field is identically zer

( ) ) 0

o

V

A dv

( A ds

→

∇ ∇× = ∇× =

∫ ⁱ ∫ ⁱ

) ˆ

of should be outward, cf a ds

1 2

ˆ ˆ

so , also outward

n

n n

a a

∫ ∫

^ˆ

∫

^ˆ

. ) ( ) ( ) ( )

0

n n

S S S

i e A ds A a ds A a ds

A dl A dl

∇× = ∇× + ∇×

= + =

∫ ∫ ∫

∫ ∫

i i i

i i

1 2

1 2

( and are same, but opposite in direction)

C C

C C

∫ ∫

2 12 Helmholtz’s Theorem 2-12 Helmholtz s Theorem

1. solenoidal and irrotational if 0 and 0 2 solenoidal but not irrotational if 0 and 0

F F

F F

∇ = ∇× =

∇ = ∇× ≠

i i

2. solenoidal but not irrotational if 0 and 0 3. irrotational but not solenoidal if 0 and 0 4. neither solenoidal nor irro

F F

F F

∇ = ∇× ≠

∇× = ∇ ≠

i

i

tational if ∇× ≠ F 0 and ∇ i F ≠ 0 4. neither solenoidal nor irrotational if ∇× ≠ F 0 and ∇ F ≠ 0

Helmholtz's theorem

: A vector field is determined to within an additive constant if

b th it di d it l ifi d h

both its divergence and its curl are specified everywhere