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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∇×∇× = ∇ ∇⋅ −∇
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