Vector Analysis
Vector Algebra
– Addition
– Subtraction
– Multiplication
Coordinate Systems
– Cartesian coordinates
– Cylindrical coordinates
– Spherical coordinates
Vector.1
Introduction
Gradient of a scalar field
Divergence of a vector field
– Divergence Theorem
Curl of a vector field
– Stoke’s Theorem
Vector.2
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.3
Scalar and Vector
Scalar field
– A scalar which is a function
of position
– Example: T=10+x
• Represented by
brightness in this picture
Vector.4
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
T = xˆ
Vector.5
Addition
Sum of two vectors
C= A+B =B+A
Graphical representation
Example
A = 2 xˆ
B = 0 . 7 xˆ + yˆ
∴ C = A + B = 2 . 7 xˆ + yˆ
Vector.6
Scalar Multiplication
Simple product
– Multiplication of a scalar
C = aB
– Direction does not change
aB
B
Vector.7
Scalar or Dot Product
A ⋅ B = AB cos θ AB
θ AB 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 θ AB is 90°, the two vectors are orthogonal
and the dot product of two orthogonal vectors is zero.
– Example:
A = 10 xˆ + 2 yˆ
B = 3 xˆ
A ⋅ B = (10 xˆ + 2 yˆ ) ⋅ 3 xˆ = 30 xˆ ⋅ xˆ + 6 yˆ ⋅ xˆ = 30
Vector.8
Vector or Cross Product
A × B = nˆ AB sin θ AB
– θ AB is the angle between the vectors
– n̂ is a unit vector normal to the plane containing the vectors
• Right-hand rule
n̂
A × B = −B × A
Vector.9
Vector or Cross Product
In cartesian coordinate system,
xˆ × yˆ = zˆ
yˆ × zˆ = xˆ
zˆ × xˆ = yˆ
xˆ
yˆ
zˆ
A × B = Ax
Bx
Ay
By
Az
Bz
Timeout
– M3.1 – 3.4
Vector.10
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.11
Cartesian Coordinates
(x,y,z)
Differential length:
d l = xˆ dx + yˆ dy + zˆ dz
Differential surface area:
Fig. 3-8
d s x = xˆ dydz
d s y = yˆ dxdz
d s z = zˆ dxdy
Differential volume:
dv = dxdydz
Vector.12
Cylindrical Coordinates
(r ,φ , z )
Vector.13
Cylindrical Coordinates
Differential length:
d l = rˆdr + φˆrd φ + zˆ dz
Differential surface area:
d s r = rˆrd φ dz
d s = φˆdrdz
φ
d s z = zˆ rd φ dr
Differential volume:
dv = rdrd φ dz
Vector.14
Example 3-4
Vector.15
Spherical Coordinates
( R ,θ ,φ )
Vector.16
Spherical Coordinates
Differential length:
d l = Rˆ dR + θˆRd θ + φˆR sin θ d φ
Differential surface area:
d s R = Rˆ R 2 sin θ d θ d φ
d s = θˆR sin θ dRd φ
θ
d s φ = φˆRdRd θ
Differential volume:
dv = R 2 sin θ dRd θ d φ
Vector.17
Example 3-5
Vector.18
Summary
Vector.19
Gradient of a Scalar Field
In Cartesian coordinate, the gradient of scalar field T is
∂f
∂f
∂f
grad f = ∇ f =
yˆ +
zˆ
xˆ +
∂x
∂y
∂z
– a vector in the direction of maximum increase of the field f.
– ∇ is an operator and defined as
∂
∂
∂
∇ ≡
xˆ +
yˆ +
zˆ
∂x
∂y
∂z
Demonstration: D3.1, D3.2, DM3.5， M3.6
Vector.20
Del Operator
The operator in cylindrical coordinates is defined as
∂
1 ∂ ˆ ∂
∇ ≡
rˆ +
φ +
zˆ
∂r
r ∂φ
∂z
In spherical coordinates, we have
∂ ˆ 1 ∂ ˆ
1
∂ ˆ
∇ ≡
R+
θ +
φ
∂R
R ∂θ
R sin θ ∂ φ
Vector.21
Divergence of a Vector Field
Divergence of a vector field A:
div A ≡ lim
∆v→ 0
∫ A ⋅ dS
S
∆v
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,
∂Ay
∂Ax
∂Az
+
+
div A = ∇ ⋅ A =
∂x
∂y
∂z
D3.10， M3.8
Vector.22
Divergence Theorem
If A is a vector, then for a volume V surrounded by a closed
surface S,
∫
V
∇ ⋅ A dv =
∫ A ⋅ dS
S
The above integral represents the net flex leaving the closed
surface S if A is the flux density
V
S
Vector.23
Curl of a Vector Field
The curl of a vector field describes the rotational property, or
the circulation of the vector field.
Examples:
Vector.24
Curl of a Vector Field
curl A ≡ ∇ × A
≡ lim
∆S → 0
nˆ ∫ A ⋅ d l
C
∆S
In Cartesian coordinates, the curl of a vector is
xˆ
∂
∇×A =
∂x
Ax
yˆ
∂
∂y
Ay
zˆ
∂
∂z
Az
Vector.25
Stoke’s Theorem
Stokes’s theorem: For an open surface S bounded by a
contour C,
∫
S
(∇ × A ) ⋅ d S =
∫ A ⋅ dl
C
C
S
The line integrals from adjacent cells cancel leaving the
only the contribution along the contour C which bounds the
surface S.
Vector.26
Exercises
Cylinder volume
Gradient
Divergence
Curl
Vector.27