Vector.1

Vector Analysis

Vector Algebra– Addition– Subtraction– Multiplication

Coordinate Systems– Cartesian coordinates– Cylindrical coordinates– Spherical coordinates

Vector.2

Introduction

Gradient of a scalar field

Divergence of a vector field– Divergence Theorem

Curl of a vector field– Stoke’s Theorem

Vector.3

Scalar and Vector

Scalar– Can be completely specified by its magnitude– Can be a complex number– Examples:

• Voltage: 2V, 2.5∠10°• Current• Impedance: 10+j20Ω

Vector.4

Scalar and Vector

Scalar field– A scalar which is a function

of position– Example: T=10+x

• Represented by brightness in this picture

Vector.5

Scalar and Vector

Vector– Specify both the magnitude and

direction of a quantity– Examples

• Velocity: 10m/s along x-axis• Electric field: y-directed

electric field with magnitude 2V/m

Vector field– Example

xT ˆ=

Vector.6

Addition

Sum of two vectors

Graphical representation

Example

ABBAC +=+=

yxyx

x

ˆˆ7.2ˆˆ7.0

ˆ2

+=+=∴+=

=

BACBA

Vector.7

Scalar Multiplication

Simple product– Multiplication of a scalar

– Direction does not change

BC a=

aB B

Vector.8

Scalar or Dot Product

is the angle between the vectors.

– The scalar product of two vectors yields a scalar whose magnitude is less than or equal to the products of the magnitude of the two vectors.

– When the angle is 90°, the two vectors are orthogonal and the dot product of two orthogonal vectors is zero.

– Example:

ABAB θcos=⋅BA

ABθ

ABθ

( ) 30ˆˆ6ˆˆ30ˆ3ˆ2ˆ10ˆ3

ˆ2ˆ10

=⋅+⋅=⋅+=⋅=

+=

xyxxxyxx

yx

BABA

Vector.9

Vector or Cross Product

– is the angle between the vectors– is a unit vector normal to the plane containing the vectors

• Right-hand rule

ABABn θsinˆ=× BA

ABθn

n

ABBA ×−=×

Vector.10

Vector or Cross Product

In cartesian coordinate system,

Timeout– M3.1 – 3.4

yxzxzyzyx

ˆˆˆˆˆˆˆˆˆ

=×=×=×

zyx

zyx

BBBAAAzyx ˆˆˆ

=× BA

Vector.11

Orthogonal Coordinate Systems

In electromagnetics, the fields are functions of space and time. A three-dimensional coordinate system allow us to uniquely specify the location of a point in space or the direction of a vector quantity.

– Cartesian (rectangular) coordinate system– Cylindrical coordinate system– Spherical

Vector.12

Cartesian Coordinates

(x,y,z)Differential length:

Differential surface area: Fig. 3-8

Differential volume:

dzzdyydxxd ˆˆˆ ++=l

dxdyzd

dxdzyddydzxd

z

y

x

ˆ

ˆˆ

=

==

sss

dxdydzdv =

Vector.13

Cylindrical Coordinates

),,( zr φ

Vector.14

Cylindrical Coordinates

Differential length:

Differential surface area:

Differential volume:

dzzrddrrd ˆˆˆ ++= φφl

drrdzddrdzd

dzrdrd

z

r

φ

φ

φ

φ

ˆ

ˆˆ

=

=

=

ss

s

dzrdrddv φ=

Vector.15

Example 3-4

Vector.16

Spherical Coordinates

),,( φθR

Vector.17

Spherical Coordinates

Differential length:

Differential surface area:

Differential volume:

φθφθθ dRRddRRd sinˆˆˆ ++=l

θφ

φθθ

φθθ

φ

θ

RdRdd

dRdRd

ddRRd R

ˆsinˆ

sinˆ 2

=

=

=

s

s

s

φθθ ddRdRdv sin2=

Vector.18

Example 3-5

Vector.19

Summary

Vector.20

Gradient of a Scalar Field

In Cartesian coordinate, the gradient of scalar field T is

– a vector in the direction of maximum increase of the field f.

– is an operator and defined as

Demonstration: D3.1, D3.2, DM3.5， M3.6

zzfy

yfx

xfff ˆˆˆ

∂∂

+∂∂

+∂∂

=∇=grad

zz

yy

xx

ˆˆˆ∂∂

+∂∂

+∂∂

≡∇

∇

Vector.21

Del Operator

The operator in cylindrical coordinates is defined as

In spherical coordinates, we have

zzr

rr

ˆˆ1ˆ∂∂

+∂∂

+∂∂

≡∇ φφ

φφθ

θθ

ˆsin1ˆ1ˆ

∂∂

+∂∂

+∂∂

≡∇RR

RR

Vector.22

Divergence of a Vector Field

Divergence of a vector field A:

v

ddiv S

v ∆

⋅≡ ∫

→∆

SAA

0lim

If we consider the vector field A as a flux density (per unit surface area), the closed surface integral represents the net flux leaving the volume ∆v

In rectangular coordinates,

zA

yA

xAdiv zyx

∂∂

+∂

∂+

∂∂

=⋅∇= AA

D3.10， M3.8

Vector.23

Divergence Theorem

If A is a vector, then for a volume V surrounded by a closed surface S,

∫ ∫ ⋅=⋅∇V S

ddv SAA

The above integral represents the net flex leaving the closed surface S if A is the flux density

V

S

Vector.24

Curl of a Vector Field

The curl of a vector field describes the rotational property, orthe circulation of the vector field.Examples:

Vector.25

Curl of a Vector Field

In Cartesian coordinates, the curl of a vector is

S

dn

curl

C

S ∆

⋅≡

×∇≡

∫→∆

lA

AA

ˆlim

0

zyx AAAzyx

zyx

∂∂

∂∂

∂∂

=×∇

ˆˆˆ

A

Vector.26

Stoke’s Theorem

Stokes’s theorem: For an open surface S bounded by a contour C,

∫ ∫ ⋅=⋅×∇S C

dd lASA )( C S

The line integrals from adjacent cells cancel leaving the only the contribution along the contour C which bounds the surface S.

Vector.27

Exercises

Cylinder volume

Gradient

Divergence

Curl