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Lecture 5 Vector Calculus Basic Concepts by nml23533

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									            ELEC3002/ELEC7005


                    Lecture 5
                Vector Calculus
                Basic Concepts

              Basic Concepts (Glyn James, Ch 7.1)
                 Introduction (7.1)
                 Basic Concepts (7.1.1)
                 Transformations (7.1.3)
Vector1/1
                                                    ITEE
      The lectures presented on Vector Calculus

 • emphasise differentiation & integral aspects.

      Applications focus on Electromagnetics.




Vector1/2
                                                   ITEE   2
                   Basic vectors (revision)

   In Electromagnetics, We are familiar (!?) with the fact that
   Electric field (E), Magnetic field (H), and Current density (J) are
   all vectors, while Potentials (), permittivity () and
   permeability () are all scalars.

  We also note that either vectors or scalars could be a function of
  position or some other variable e.g. time, frequency. So we could
  write
                                                    we could have used
                            
                            E  x, y , z , t 
The arrow means “vector”,                           another coordinate
sometimes written as bold
                                                    system instead!


   Vector1/3
                                                              ITEE       3
                       Vector point function
 Graphically, we use the arrow tipped line to represent the vector

                                                            
                                 AA                         A  Aa
                                                                  ˆ
magnitude of                                                                     
                                                                                 A
a unit vector is 1                                    we can always           ˆ 
                                                                              a
                                                      construct a unit vector    A
     or a  1
        ˆ
                             ˆ
                             a
                                       unit vector                         
                                                                 x  a x  e x  i,
                                                                 ˆ ˆ
                                                                           
               For a Cartesian system of coordinates we          y  a y  e y  j,
                                                                 ˆ ˆ
               use the following alternative notations for                 
                                                                 z  a z  ez  k
                                                                 ˆ ˆ
               base vectors:
   Vector1/4
                                                                              ITEE    4
We can write any vector in terms of their base vectors:
                  
          
                  A  Ax x  Ay y  Az z
                          ˆ     ˆ      ˆ
       x  a x  e x  i,
       ˆ ˆ
                                             x, y, and z components
       y  a y  e y  j,
       ˆ ˆ                                                   
                                             of the vector A
       ˆ  a z  ez  k
       z   ˆ

                                                 
The magnitude of the vector then becomes         A  Ax2  Ay  Az2
                                                            2



  Keep in mind that generally each of the components of the
  vector may still be a function of x,y,z (or t or ) as well. ie.
         
         A  Ax  x, y, z  x  Ay  x, y, z  y  Az  x, y, z  z
                            ˆ                  ˆ                  ˆ

Vector1/5
                                                              ITEE     5
                   Position Vector

 In a Cartesian system, a position vector r is a vector from the
 origin to a point (x,y,z).
 This is especially useful for coordinate references.
                               z

                                        
                                        r  xx  yy  zz
                                             ˆ    ˆ ˆ

                                              y
               x


Vector1/6
                                                          ITEE     6
            addition & subtraction of vectors

                                                              
                                                              R1  x1 x  y1 y  z1 z
                                                                      ˆ      ˆ      ˆ
                                                              
                                                              R2  x2 x  y2 y  z2 z
                                                                       ˆ      ˆ       ˆ
                                                             the vector R12 is the vector from
                                                              P1 to P2 and its distance (length
                                                             or magnitude) is d:

                                                        
                                             R12  R2  R1
          
      d  R12
                                                  x  x2  x1   y  y2  y1   z z2  z1 
                                                   ˆ               ˆ               ˆ

              
              x2  x1    y2  y1    z 2  z1 
                         2             2             2 12
                                                         
Vector1/7
                                                                                     ITEE         7
            Vector addition



            C  x  Ax  Bx   y ( Ay  B y )  z ( Az  Bz )
                ˆ               ˆ                ˆ


             Subtraction is equivalent to the addition of
             A to negative B. ie. D = A – B = A + (-B)




Vector1/8
                                                            ITEE   8
                Dot Product (scalar product)

                                                        Notice the similarity
                          
                       A  B  A B cos AB 
                                                        of the vector projection
                                                        and the more general
                                                        matrix algebra projection
                                                        onto subspaces!


                                                  AB     is a smaller angle
                                                          between A and B.


•       Always yields a scalar!
•       A cos(AB) is the component of A along B.
        We say this is the projection of A on B.
•       If two vectors are orthogonal their dot product is zero ie x  y  0
                                                                   ˆ ˆ
•       A·A=|A|2=A2

    Vector1/9
                                                                       ITEE         9
                    two definitions
Note also that if two vectors are:
                        
                        A  a1i  a2 ˆ  a3 k
                               ˆ      j      ˆ
                        
                        B  b1i  b2 ˆ  b3 k
                              ˆ      j      ˆ

Then the dot product can be written either as:
                      
                   A  B  A B cos  AB
                    
                   A  B  a1b1  a2b2  a3b3

  It can be easily shown that these two definitions are equivalent
  if the cosine of the included angle is expressed in terms of the
  direction cosines of the two vectors.
Vector1/10
                                                          ITEE       10
                   Scalar/dot product
Commutative law:
                      
                   A B  B  A

Distributive law:
                            
             A  ( B  C)  A  B  A  C




Vector1/11
                                            ITEE   11
             Law of cosines for a triangle
The law of cosines is a scalar relationship that expresses the
length of a side of a triangle in terms of the lengths of the two
other sides and the angle between them:

                            C  A2  B2  2 AB cos AB
               C=B-A



                                             
  C  C  C  ( B  A)  ( B  A)  B  B  A  A  2 A  B 
     2


          A2  B 2  2 AB cos AB

Vector1/12
                                                            ITEE    12
         Cross product (vector product)
                       Note how the cross product produces
                       the normal to the plane containing the
                       two vectors! This will relate tangential
                       and normal components.



                          !!!!Important!!!

                       Alternatively, the screwdriver rule:
                       Rotating A to B moves a
                       screw in the resultant direction




Vector1/13
                                                              ITEE   13
                Vector/Cross product
The cross product is not commutative:
                         
                 A  B  B  A

The cross product is not associative:

                             
             A  ( B  C)  ( A  B)  C




Vector1/14
                                           ITEE   14
                   Cross product (ctd)

        ˆ
        x                  Move clockwise in the direction of the arrow
                           the cross product is positive. Move in the counter-clockwise
                           direction and the cross product is negative.

ˆ
z             ˆ
              y                         x y  z
                                        ˆ ˆ ˆ
                                        zx  y
                                        ˆ ˆ ˆ
                                        x z  y
                                        ˆ ˆ      ˆ
           xˆ      yˆ     zˆ
                                              note that the cross product
   A  B  Ax      Ay     Az   B  A            define the right handed coordinate
           Bx      By     Bz                      system

      x( Ay Bz  Az B y )  y ( Az Bx  Ax Bz )  z ( Ax B y  Ay Bx )
       ˆ                     ˆ                     ˆ
Vector1/15
                                                                           ITEE        15
                         Application
In mechanics, the moment m of a force P about a point Q is defined
         
as m  P d where d is the (perpendicular) distance between Q and
the line of action L of P. any vector from
                            Q to any point                L
                            A on L
                                           P
                      Q      r                    we can now define
      then                                       a moment vector
           
       d  r sin 
                                                  m, its direction is
                      d                A          perpendicular to
                                                  both r and p
     so
                                                 
       
 m  r p sin       or
                              
                          m rp               mrp
 Vector1/16
                                                            ITEE   16
                  Another example
In electromagnetics we deal with the flow of power by an EM wave.
For any wave with an electric field E and a magnetic field H, we
define a Poynting vector S given by
                        
                      S  EH        W/m 2  
The direction of S is along the propagation direction (k), thus S
represents the power per unit area (power density) carried by the
wave. If the wave is incident on an aperture of area A with an
outward unit surface vector n, then the total power intercepted by
the aperture is                                         nˆ
                 
           P   S  n dA  W 
                     ˆ                                   kˆ
               A
                                                  A
   scalar!                             S

  Vector1/17
                                                           ITEE      17
             Product of Three Vectors
The scalar triple product:
                             
    A  ( B  C)  B  (C  A)  C  ( A  B)
                                     Note the cyclic permutation of the order
                                     of the three vectors A, B and C.



 The vector triple product:
                          
         A  ( B  C)  B( A  C)  C( A  B)
                                     “BAC-CAB” rule.


Vector1/18
                                                                     ITEE       18
       Coordinate systems: Cylindrical
   We commonly deal with the (x,y,z) –Cartesian coordinates.
   We sometimes have to deal with other orthogonal coordinate systems,
   for example Cylindrical or Spherical systems.
                                                        unit vectors
                                z    Cylindrical        are
                                                              , , z
                                                             ˆ ˆ  ˆ
 Note that  is
 always measured                           , , z 
 in the x-y plane                                             z
                                                           ˆ ˆ ˆ
 perpendicular to z                        z
 and measured from the                                     z  
                                                           ˆ ˆ ˆ
 x-axis!                               
                     x                         y          z 
                                                           ˆ ˆ ˆ

A vector in a cylindrical coordinates is written as:
                  
                     A   A   A  z Az
                         ˆ       ˆ     ˆ
Vector1/19
                                                                  ITEE   19
             Coordinate systems: Spherical

                  Spherical    z
 Again:                                   r , ,  
 r   
 ˆ ˆ ˆ
                                   r
                                                    = azimuth
                                                    = elevation

                           
          x                                        y
A vector in a spherical coordinates is written as:
                    
                    A  r Ar   A   A
                        ˆ       ˆ      ˆ
Vector1/20
                                                            ITEE   20
             Transformation of Coordinates
Occasionally we need to transform between different coordinate systems.
Transformations of the position vector are quite simple.

Some of the programming environment like Matlab can perform
transformations between large number of systems.

[THETA, PHI,R] = cart2sph(X,Y,Z) transforms Cartesian coordinates
stored in corresponding elements of arrays X,Y and Z into spherical
Coordinates. Note here that azimuth Theta and Elevation Phi are in radians.



 cart2pol Transform Cartesian coordinates to polar or cylindrical
 pol2cart Transform polar or cylindrical coordinates to Cartesian
 sph2cart Transform spherical coordinates to Cartesian

Vector1/21
                                                                    ITEE      21
             Transformation of Coordinates
Cylindrical to Cartesian    Cartesian to Cylindrical
    x   cos                     x2  y 2
    y   sin                          y
                                  tan   1
    zz                                 x
                                zz
                              Cartesian to Spherical
Spherical to Cartesian
                                 R  x2  y 2  z 2
    x  R sin cos
                                                   x2  y 2
    y  R sin sin               tan   1

                                                     z
    z  R cos                                 y
                                  tan1
                                               x
Vector1/22
                                                              ITEE   22
      Jacobian matrix (James sect 7.1.3)
 The derivative of a coordinate transformation is the matrix of its partial
 derivatives. In the case of 3D coordinate systems this is always a “three by
 three” matrix. This matrix is sometimes called the Jacobian matrix.
 James considers the 2D Jacobian matrix that transforms derivatives between
 the (x,y) plane and the (s,t) plane.
           s  s(x,y)               u u s u        t
                                                             The chain rule
                                    x s x t        x
           t  t(x,y)               u u s u        t
                                                               application may be
                                                             regarded as a
               u  f ( x, y )       y s y t        y      transformation
becomes
               u  F ( s, t )
                                    u   s   t   u 
                                    x   x   x   s 
 u  F ( s ( x, y ), t ( x, y ))                 
                                    u   s   t   u 
                                     
                                    y   y       
                                                 y   t 

  Vector1/23
                                                                              ITEE   23
 Jacobian determinant (James sect 7.1.3)
The determinant of this matrix is called the Jacobian determinant
of the transformation, or else just the Jacobian.
                         s t
                                               useful for implementing
             ( s, t ) x x s x t x
        J              s t                changes in variables in
             ( x, y )            sy t y       surface & volume integrals
                         y y
This determinant measures how infinitesimal volumes change under the
transformation. For this reason, the Jacobian determinant is the multiplicative
factor needed to adjust the differential volume form when you change
coordinates. Jacobian of the inverse transformation is
                    x    y                       ( x, y )  ( s , t )
         ( x, y ) s                                                   1
   J1                   s  xs     ys
                                                   ( s , t )  ( x, y )
         ( s, t ) x     y xt       yt
                    t    t                     or       J1  J 1
Vector1/24
                                                                          ITEE    24
        Jacobian matrix and determinant
                       Transformation from x-y     s  2x  y
Example:               plane to s-t plane
                                                   t  x  2y
                s     t
                x     x 2 1
             J                 5
                s     t 1  2
                                                      1
                y     y                          x  (2s  t )
                                                      5
      If we now invert the transformation             1
                                                   y  ( s  2t )
                  x   y 2 1                         5
                  s   s   5 5    1     Note that if J=J1 =0, then the
             J1               
                  x   y 1 2      5     variables are functionally dependent
                             
                  t   t   5 5          and imply a non-unique
                                         correspondence between the
      Confirming that J1=J-1             (x,y) and (s,t) planes
Vector1/25
                                                                     ITEE       25
 Jacobian determinant (James sect 7.1.3)
The definition of a Jacobian is not restricted to functions of two variables. For
example, for functions of three variables:
              u  U ( x, y, z ); v  V ( x, y, z ); w  W ( x, y, z );
 the Jacobian matrix and its determinant is obtained from :


                               u          v     w
                               x          x     x    u x v x wx
                 (u , v, w) u            v     w
             J                                       u y v y wy
                 ( x, y, z ) y           y      y
                                                        u z v z wz
                               u          v     w
                               z          z      z

Vector1/26
                                                                         ITEE       26
               Jacobian determinant
Example:        Show that if x+y=u and y=uv

                              ( x, y )
                          J            u
                              (u , v)


  Solution:
                      x  u  y  u  uv
                      y  uv

                             x y
                  ( x, y ) u u 1  v v
              J                         (1  v)u  uv  u
                  (u, v) x y      u u
                             v v
Vector1/27
                                                                ITEE   27
             Jacobian determinant
Example:         x  e u cos v          y  e u sin v
                                 ( x, y )                        (u , v)
             Determine :     J                 and       J1 
                                 (u, v)                          ( x, y )
              And verify that they are mutual inverses.
                                                                x y
Solution:                                            ( x, y ) u u
                                                 J                   e 2u
                                                     (u, v) x y
                                                                v v

                                                                     (u , v)
                                                             J1               
                                                                     ( x, y )
                                                                         1       1
                                                                    2          2u
                                                                    x  y2 e
                                 J  J1  1
Vector1/28
                                                                       ITEE     28
                             Curves in space
The parametric representation of a curve in space is the most useful.
Basically, the point in space (x,y,z) is a function of a parameter t.
                  x  x(t ), y  y (t ), z  z (t )
We let t vary over some range and the points trace out a curve.
To put it in a vectorial form we draw a position vector to the point
on the curve C.
                                          z                 C
       Note that this gives
                                              r(t)
              some orientation to
              the curve +ve/- depending
              on whether the parameter t is   x   y
              increasing or decreasing.


 Vector1/29
                                                          ITEE     29
                     Some examples
For example a helix:         r  t   6 cos t x  6sin t y  t z
                                               ˆ          ˆ     ˆ

                       Point moves in a circular fashion in x-y and then along the z direction:

A straight line through a point A with a position vector a in the
direction of a constant vector b is:
                                       
                                                               
                 z                     bA
                                                    r  t   a  tb
                            
                            a
             x          y


Vector1/30
                                                                             ITEE         30
                          unit tangents
A tangent (vector) to the curve is given by the derivative of the
curve vector wrt the parameter.

e.g. A tangent vector to the helix r  t   6 cos t x  6sin t y  t z is given by :
                                                     ˆ          ˆ     ˆ

                   
                   r   t   6sin t x  6 cos t y  z
                                       ˆ           ˆ ˆ

 A unit tangent vector is obtained from the tangent vector by
 normalization (ie dividing r'(t) by its length).
                             
                          r              Tangents are very useful,
                        u                we use them to approximate
                             r            curves with straight lines!



Vector1/31
                                                                         ITEE       31
                                            Arc Length
  The arc length along a curve is also a useful quantity. We define
  this in terms of the magnitude of the derivative:
                                            b                  b
       length                                                 
       variable s              s (t )         r   r dt   r  dt
                                            a                  a

                        
   Example: for a curve r (t )  t 2 x  t 3 y
                                     ˆ       ˆ
                                                         
r ' (t )  2t x  3t 2 y  t (2 x  3ty )
              ˆ        ˆ        ˆ     ˆ         r ' (t )  r ' (t )  (2 x  3ty )  (2 x  3ty )  t 4  9t 2
                                                                         ˆ     ˆ        ˆ     ˆ

  we can show that the length of the curve starting at the point t = 0
  to some point p is:             p

                        s  t    t 4  9t 2 dt
                                                   0
  Vector1/32
                                                                                                ITEE         32
                Arc length as a parameter

Instead of using the parameter t along the curve we can use the
arc length s instead! If we do this, the tangent vector r'(s) is the
unit tangent vector.
                                                                    Note the
  It is reasonably easy to parameterise a curve in terms           variable ‘s’
  of the arc length:
    1.       Get s(t) by computing the arc length integral from 0 to t.
    2.       Invert the function s =s(t) to t = t(s)
    3.       Write x = x(s) = x(t(s)), y = y(s) = y(t(s)), to obtain r(s)
    4.       The magnitude of the tangent will be 1. Prove this by
             taking the derivative.

Vector1/33
                                                                  ITEE        33
             Tangent, normal and binormal

 There are other vectors associated with curves such as a normal
  to the curve at a point and a binormal vector that forms a 3D
 coordinate system along the tangent vector at any point along the
 curve.
                          normal              These are used mainly
                                              in dynamics and optics
                                              giving rise to concepts of
                z tangent                     velocity acceleration,
                                              curvature and torsion etc.
                             binormal

        x            y


Vector1/34
                                                              ITEE         34

								
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