Vector Analysis. Vector Algebra Addition Subtraction Multiplication

Transcription

1 Vector nalsis Vector lgebra ddition Subtraction Multiplication Coordinate Sstems Cartesian coordinates Clindrical coordinates Spherical coordinates Vector.1

2 Introduction Gradient of a scalar field Divergence of a vector field Divergence Theorem Curl of a vector field Stoke s Theorem Vector.2

3 Scalar and Vector Scalar Can be completel specified b its magnitude Can be a comple number Eamples: Voltage: 2V, Current Impedance: 10j20Ω Vector.3

4 Scalar field scalar which is a function of position Eample: T10 Represented b brightness in this picture Scalar and Vector Vector.4

5 Vector Specif both the magnitude and direction of a quantit Eamples Velocit: 10m/s along -ais Electric field: -directed electric field with magnitude 2V/m Vector field Eample T Scalar and Vector Vector.5

6 ddition Sum of two vectors C B B Graphical representation Eample 2 B 0.7 C B 2.7 Vector.6

7 Simple product Multiplication of a scalar C ab Scalar Multiplication Direction does not change B ab Vector.7

8 Scalar or Dot Product B B cos θ B θ B is the angle between the vectors. The scalar product of two vectors ields a scalar whose magnitude is less than or equal to the products of the magnitude of the two vectors. When the angle θ B is 90, the two vectors are orthogonal and the dot product of two orthogonal vectors is ero. Eample: B 10 3 B 2 ( 10 2 ) Vector.8

9 Vector or Cross Product B n B sin θ B θ B is the angle between the vectors n is a unit vector normal to the plane containing the vectors Right-hand rule n B B Vector.9

10 Vector.10 Vector or Cross Product In cartesian coordinate sstem, Timeout M B B B B

11 Orthogonal Coordinate Sstems In electromagnetics, the fields are functions of space and time. three-dimensional coordinate sstem allow us to uniquel specif the location of a point in space or the direction of a vector quantit. Cartesian (rectangular) coordinate sstem Clindrical coordinate sstem Spherical Vector.11

12 Cartesian Coordinates (,,) Differential length: dl d d d Differential surface area: Fig. 3-8 ds ds ds dd dd dd Differential volume: dv ddd Vector.12

13 Clindrical Coordinates ( r, φ, ) Vector.13

14 Clindrical Coordinates Differential length: dl rdr φrd φ d Differential surface area: ds rrd φd ds ds r φ φdrd rd φdr Differential volume: dv rdrd φd Vector.14

15 Eample 3-4 Vector.15

16 Spherical Coordinates ( R, θ, φ ) Vector.16

17 Spherical Coordinates Differential length: dl RdR θ Rd θ φr Differential surface area: RR 2 s sin θdθd d R ds ds θ φ θ R sin θdrd φ φrdrd θ φ sin θdφ Differential volume: dv R 2 sin θdrd θdφ Vector.17

18 Eample 3-5 Vector.18

19 Summar Vector.19

20 Vector.20 Gradient of a Scalar Field In Cartesian coordinate, the gradient of scalar field T is a vector in the direction of maimum increase of the field f. is an operator and defined as Demonstration: D3.1, D3.2, DM3.5, M3.6 f f f f f grad

21 Vector.21 Del Operator The operator in clindrical coordinates is defined as In spherical coordinates, we have r r r 1 φ φ φ φ θ θ θ sin 1 1 R R R R

22 Divergence of a Vector Field Divergence of a vector field : div lim v 0 S d v S If we consider the vector field as a flu densit (per unit surface area), the closed surface integral represents the net flu leaving the volume v In rectangular coordinates, div D3.10, M3.8 Vector.22

23 Divergence Theorem If is a vector, then for a volume V surrounded b a closed surface S, dv V S d S The above integral represents the net fle leaving the closed surface S if is the flu densit V S Vector.23

24 Curl of a Vector Field The curl of a vector field describes the rotational propert, or the circulation of the vector field. Eamples: Vector.24

25 Vector.25 Curl of a Vector Field In Cartesian coordinates, the curl of a vector is S d n curl C S l lim 0

26 Stoke s Theorem Stokes s theorem: For an open surface S bounded b a contour C, ( ) d S S C C d l S The line integrals from adjacent cells cancel leaving the onl the contribution along the contour C which bounds the surface S. Vector.26

27 Eercises Clinder volume Gradient Divergence Curl Vector.27