Vector Analysis. Vector Algebra Addition Subtraction Multiplication

Vector Analysis Vector Algebra – Addition – Subtraction – Multiplication Coordinate Systems – Cartesian coordinates – Cylindrical coordinates – Spheri...
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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 – nˆ is a unit vector normal to the plane containing the vectors • Right-hand rule



A × B = −B × A Vector.9

Vector or Cross Product In cartesian coordinate system,

xˆ × yˆ = zˆ yˆ × zˆ = xˆ zˆ × xˆ = yˆ xˆ





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