Review: Vector Algebra And Analysis

R1.1 Scalar ProductR1.2 Vector ProductR1.3 Gradient and Del operatorsR1.4 DivergenceR1.5 Curl of a Vector FieldR1.6 Laplacian OperatorR1.7 Other Operators

x

2 2 2 2 2 2x x

For two vectors:

a=a

b= k

and θ is the angle between a and b.

The magnitudes of a and b are:

a

Scalar and Vector Product

= a , b= ,

s

y z

x y z

y z y z

i a j a k

b i b j b

a a b b b

+ +

+ +

+ + + +

R1.1

x

a b=abcosθ (1.1)

a b=a (1.2)x y y z zb a b a b

⋅

⋅ + +

Scalar product

ab

θ

a b =absinθ (2.1)

a b= - b a

The direc

tion of a b is determined by the Right-Hande-

(2.1

Rule.

)

a b=(

×

×

× ×

×

R1.2 Vector product

x

y z y z x z x y x

a ) ( k)

=(a -a b ) (a -a b )j+ (a -a b )k (2.3)

The above equation can also be expressed as :

y z x y z

z x y

x y z

x y z

i a j a k b i b j b

b i b b

i j ka b a a a

b b b

+ + × + +

+

× = (2.4)

1.3 Gradient (Grad) and del Operators

Fig.3 Two close contours in a scalar field in the x-ycoordinate system.

is a Vector Operator:

i j k

=( i+ j+ k)x y z

= i+ j+ kx y z

The gradient is the maximum rate of change

G

of t

rad =

x y z

φ

φ

φ

φ

φ

φ

∇∂ ∂ ∂

∇ = + +∂ ∂ ∂

∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂

∇

∂

Gradient (Grad) and del Operators

he scale field , which is perpendicular to the contour at that point.

Physical Example: Electric potential and electric field.

φ

1.4 Divergence (Div)

Fig. 5. The relationship of V and the

small vector area A

Fig. 6. The cube box surrounding the

point P, and the vector field V.

=( i+ j+ k) ( )x y z

= + + (4.6)x x x

The divergence is the outward flux per unit volume

Div =

at tha

x y z

yx z

V i V j V k

VV V

V V ∂ ∂ ∂⋅ + +

∂ ∂ ∂∂∂ ∂

∂ ∂ ∂

∇ ⋅

Divergence (Div)

t point. It is a scalar quantity.

Physical Example: electric flux per unit volume.

Sv VV d

Diver

s=

gence T

(

heorem

)V dv⋅ ∇ ⋅∫ ∫

1.5 Curl of a Vector Field

Fig. 6. The circulation in a square contour in the x-y plane.

=

=( - ) i+( - )j+( - )k (5.2)

The Z compon

Cu

ents of curl V is the circulatio

rl

n

=

x y z

y yx xz z

i j k

x y zV V V

V VV VV Vy z z x

V V

x y

∂ ∂ ∂∂ ∂ ∂

∂ ∂∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂

∇×

Curl of a Vector Field

cV ds per unit area in the x-y plane.

In a x-y-z coordinate, the vector field V has a rotatory component in a plane whose

normal is in the direction of V. The circulation per unit area in that pl

⋅∫

ane is given by

V∇×

2 2 2

2 2

2

2

Laplacian operator is a scalar operator, and is defined as :

= (6.1)x

For a vector field V x y z

y z

V i V j V k

∂ ∂

∇ =

∂+ +

∂ ∂ ∂

+

∇⋅∇

+

1.6 Laplacian Operator

1.7 Other opera o

=

t rs

2

, we can prove that

( ) (7.1)V V V∇×∇× = ∇ ∇⋅ −∇