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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