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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.

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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

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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:
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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

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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

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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

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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)

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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

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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.
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Scalar/dot product
Commutative law:
   
A B  B  A

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

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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

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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

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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

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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
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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
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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

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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.

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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
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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
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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

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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
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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 

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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
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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
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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

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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
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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
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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.

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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

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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!

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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
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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.

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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
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