Chapter 2 Vector Analysis

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Chapter 2 Vector Analysis

Spatial derivative of Scalar & Vector field:- Gradient of a Scalar field- Divergence of a vector field

Recall: given E, to find V (from vector to scalar)Now, what if given scalar field V, can we find a vector field E ???

RV E d r

∞= − ⋅∫

2-5 Gradient of a Scalar Field

Using gradient of a Scalar Field

2 equal-potential surfaces, V1, V1+dV3 points: P1 at surface V1; P2, P3 at same surface V1+dVP1P2: along normal direction an

P1P3: al · an = cos θ

( )dV V dl= ∇ ⋅


grad ndVV V adn

= ∇

also have

Definition of gradient of a scalar field:

We define the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar as the gradient of that scalar.

cos n l ldV dV dn dV dV a a V adl dn dl dn dn

α≡ = = ⋅ = ∇ ⋅

Space rate of increase in the al direction: The projection of ∇V in the direction of al

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In Cartesian coordinates:

x y za a ax y z∂ ∂ ∂

∇ ≡ + +∂ ∂ ∂


x y zV a a a Vx y z

⎛ ⎞∂ ∂ ∂∇ = + +⎜ ⎟∂ ∂ ∂⎝ ⎠

Vector differential operator:

Example 2-9: The electrostatic field intensity E is derivable as the negative gradient of an scalar electric potential V; that is, E = - ∇V. Determine E at the point (1, 1, 0) if

(a)

(b)0

0

sin4

cos

x yV V e

V E R

π

θ

−=

=

Vector differential operator in general orthogonal curvilinear coordinators :

1 2 3

1 1 2 2 3 3

u u ua a ah u h u h u

⎛ ⎞∂ ∂ ∂∇ ≡ + +⎜ ⎟∂ ∂ ∂⎝ ⎠

Example 2-9: The electrostatic field intensity E is derivable as the negative gradient of an scalar electric potential V; that is, E = - ∇V. Determine E at the point (1, 1, 0) if

(a)

(b)0

0

sin4

cos

x yV V e

V E R

π

θ

−=

=

3


In other coordinates: (see last page in the book)

Recall Gauss’s law :

2-6 Divergence of a Vector Field

For closed surface (sphere), net-electric-flux :

oS

E d S Qε ⋅ =∫

S

E d SΦ = ⋅∫

Div. is a measure of the strength of a flow source.A net outward flux of a vector A through a surface bounding a volume indicate the present of a flow source.

Divergence in Cartesian :

0div lim S

V

A d SA A

V→

⋅= ∇ ⋅ ∫

Divergence of a vector A : Net outward flux of A per unit volume as the volume about the point tends to zero

div yx zAA AA A

x y z∂∂ ∂

= ∇ ⋅ = + +∂ ∂ ∂

How do we find it ?


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div yx zAA AA A

x y z∂∂ ∂

= ∇⋅ = + +∂ ∂ ∂We wish to derive ∇·A @ point P(x0, y0, z0) :

- Consider a differential volume with side Δx, Δ y, Δ z- Center P(x0, y0, z0)- 6 surfaces (f, back, r, l, t, bottom)


∇·A in general orthogonal curvilinear coordinators :

( ) ( ) ( )2 3 1 1 3 2 1 2 31 2 3 1 2 3

1A h h A h h A h h Ah h h u u u

⎡ ⎤∂ ∂ ∂∇ ⋅ = + +⎢ ⎥∂ ∂ ∂⎣ ⎦

See the last page in textbook for ∇·A in Cylindrical and Spherical coordinators

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Example 2-10: p47Find the divergence of the position vector to an arbitrary point.

Example 2-11: p47The magnetic flux density B outside a very long current-carrying wire is

circumferential and is inversely proportional to the distance to the axis of the wire. Find ∇·B

1) For small differential volume :

2) For any arbitrary volume V:

2-7 Divergence Theorem

3) Internal surfaces : cancel each other (due to opposite direction of dS)

4) External surfaces : net contribution, due to external surface S bounding the volume V

0div lim S

V

A d SA A

V→

⋅= ∇ ⋅ ∫Definition of divergence:

( )j

j SjA V A d S∇⋅ = ⋅∫

( )0 01 1

lim limjj j

N N

j SV Vjj jA V A d S

→ →= =

∇ ⋅ = ⋅∑ ∑ ∫

Div A at a point is defined as the net outward flux of A per unit volume as the volume about the point tends to zero

Divergence theorem :Significance: It transformed the volume integral of the divergence of a vector field to a closed surface integral of the vector field, and vice versa

How to derive ?

Divergence theorem :

(sum up)

( )V S

A dV A d S∇⋅ = ⋅∫ ∫

( )V S

A dV A d S∇⋅ = ⋅∫ ∫

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2-7 Divergence Theorem

Significance of the divergence theorem: It converts a volume integral of the divergence of a vector to a closed surface integral of the vector, and vice versa.

Example 2-12: Given a vector , verify the divergence theorem over a cube one unit on each side. The cube is situated in the first octant of the Cartesian coordinate system with one corner at the origin.

2x y zA a x a xy a yz= + +

( )V S

A dV A d S∇⋅ = ⋅∫ ∫

( )V S

A dV A d S∇⋅ = ⋅∫ ∫2

x y zA a x a xy a yz= + +

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There are two types of sources:

2-8 Curl of a vector field

Curl of a vector A is defined as max net circulation of vector A per unit area as area 0

0A∇⋅ ≠Flow source : div A is a measure of the strength of the flow source

Vortex source : curl of A is a measure of the strength of the vortex source0A∇× ≠

Vortex source causes a circulation of a vector field around it.

Net circulation: circulation of A around contour C (2 84)C

A dl⋅ −∫Curl of a vector A (Definition):

max

0lim

nC

s

a A dlA

s→

⎡ ⎤⋅⎣ ⎦∇×∫


x y z

x y z

a a a

Ax y z

A A A

∂ ∂ ∂∇× =

∂ ∂ ∂

We wish to derive ∇×A @ point P(x0, y0, z0) :- we can calculate x-components first (∇×A)x

- Consider a differential rectangular area Δ y, Δ z- Center P(x0, y0, z0)- 4 sides (1, 2, 3, 4)

∇×A in Cartesian :

∇×A in general orthogonal curvilinear coordinators :

31 21 2 3

1 2 3 1 2 3

1 1 2 2 3 3

1u u ua h a h a h

Ah h h u u u

h A h A h A

∂ ∂ ∂∇× =

∂ ∂ ∂

See the last page in textbook for ∇×A in Cylindrical and Spherical coordinators

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Example 2-14: Given a vector , find its circulation around the path OABO shown in fig. 2.22.

2x yF a xy a x= −

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2-8 Curl of a vector fieldExample 2-15:

Show that if (a) in cylindrical coordinates, where k is a constant, or(b) in spherical coordinates, where f (R) is any function of the

radial distance R

0A∇× =

( / )A a k rφ=( )RA a f R=

1) For small differential area :

2) For any arbitrary surface S :

3) Internal contour : cancel each other ( due to opposite directions of dl )

4) External contour : net contribution, due to external contour C bounding the entire area S

Definition of curl of a vector:

( ) ( ) ,j

j j n jCjA S A dl S a S∇× ⋅ = ⋅ = ⋅∫

( ) ( )0 01 1

lim limjj j

N N

j CS Sjj jA S A dl

→ →= =

∇× ⋅ = ⋅∑ ∑ ∫

How to derive ?

2-9 Stokes’s Theorem

Stokes’s theorem : Significance: It transforms the surface integral of the curl of a vector field over a open surface to a closed line integral of the vector field over the contour bounding the surface, and vice versa

Curl of a vector A is defined as max net circulation of vector A per unit area as area 0

max

0lim

nC

s

a A dlA

s→

⎡ ⎤⋅⎣ ⎦∇×∫

C

(sum up)

Stokes’s theorem :

( )S C

A d S A dl∇× ⋅ = ⋅∫ ∫

( )S C

A d S A dl∇× ⋅ = ⋅∫ ∫

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Significance of Stokes’s theorem: It transforms the surface integral of the curl of a vector field over a open surface to a closed line integral of the vector field over the contour bounding the surface, and vice versa

Special case 1: Surface integral of ∇×A is carried over closed surface (3D)

( ) 0S

A d S∇× ⋅ =∫

Special case 2: 3D 2D, such as 2D disk

What about the direction of dS and dl ?

The relative directions of dl and dS (dan) follow the right-hand rule !

( )S C

A d S A dl∇× ⋅ = ⋅∫ ∫


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2-10 Two Null Identities

Identities I

0V∇×∇ =

Identities I :

Two identities involving “del” operation: ( ) 0A∇⋅ ∇× =

Identities II

The curl of the gradient of any scalar field is identically zero.0V∇×∇ =

How to derive ? By using Stokes’s theorem

( ) 0S C C

V d S V dl dV∇×∇ ⋅ = ∇ ⋅ = =∫ ∫ ∫

Example: If , then we can define electric scalar potential V:0E∇× = E V= −∇

Converse statement : If a vector field is curl-free, then it is a conservative (or irrotational) field, and can be expressed as the gradient of a scalar field.

( )S C

A d S A dl∇× ⋅ = ⋅∫ ∫

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( ) 0A∇⋅ ∇× =Identities II : The divergence of the curl of any vector field is identically zero.

2-10 Two Null Identities

How to derive ? By using Divergence theorem

Example: If , then we can define magnetic vector potential A:0B∇⋅ = B A= ∇×

Converse statement : If a vector field is divergenceless, then it is solenoidal field and can be expressed as the curl of another vector field.

( )V S

A dV A d S∇⋅ = ⋅∫ ∫

( )S C

A d S A dl∇× ⋅ = ⋅∫ ∫

( )( ) ( )V S

A dV A d S∇⋅ ∇× = ∇× ⋅∫ ∫Note: The right side of above equation, is a closed surface. We may split it into 2 open surfaces so as to use Stokes’s theorem.

1 2 1 2

1 2( ) ( ) ( ) 0n n

S S S C C

A d S A a dS A a dS A dl A dl∇× ⋅ = ∇× ⋅ + ∇× ⋅ = ⋅ + ⋅ =∫ ∫ ∫ ∫ ∫

Summary:

Reviewed the basic rules of vector addition and subtraction, and of products of vectorsExplained the properties of Cartesian, cylindrical, and spherical coordinate systemsIntroduced the differential del ( ) operator, and defined the gradient of a scalar field, and the divergence and the curl of a vector fieldPresented the divergence theorem that transformed the volume integral of the divergence of a vector field to a closed surface integral of the vector field, and vice versaPresented the Stokes’s theorem that transforms the surface integral of the curl of a vector field to a closed line integral of the vector field, and vice versaIntroduced two important null identities in vector field


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Chapter 2 Vector Analysis