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Position Vector and Coordinate Systems Differentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)

Lecture 2: Position Vector and Coordinate Systems (cont.)

Dr.-Ing. Rene Marklein (University of Kassel) Mathematical Foundation of EFT (MFEFT) WS 2007/2008 22 / 71
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MFEFT - Lecture 2

1 Introduction

2

Vector and Tensor Algebra3 Position Vector and Coordinate Systems

Cartesian CoordinatesEinsteins Summation ConventionDifferentiation of the Position Vector; Kronecker Symbol (Kronecker Delta)Cylinder Coordinate SystemOrthogonal Curvilinear Coordinate SystemSpherical Coordinate SystemDupin Coordinates

4 Vectors: Scalar Product; Vector Product; Dyadic Product

5 Vector and Tensor Analysis

6 Distributions

7 Complex Analysis

8 Special Functions

9 Fourier Transform

10 Laplace Transform

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Cartesian Coordinate SystemPosition Unit Vector

Position Unit VectorThe Unit Vector of the Position Vector R is R (Unit Position Vector):

R =R

R(37)

=xex + yey + zezx2 + y2 + z2 . (38)

If we build R/xi exi for i = 1, 2, 3, it follows with the summation convention the form

R

xiexi

= R (39)

= R(40)

with the so-called Nabla Operator

=

xex +

yey +

zez =

3i=1

xiexi

=

xiexi

. (41)


( )

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R =R

R(37)

=xex + yey + zezx2 + y2 + z2 . (38)


R

xiexi

= R (39)

= R (40)


=

xex +

yey +

zez =

3i=1

xiexi

=

xiexi

. (41)


P i i V d C di S Diff i i f h P i i V K k S b l (K k D l )

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R =R

R(37)

=xex + yey + zezx2 + y2 + z2 . (38)


R

xiexi

= R (39)

= R (40)


=

xex +

yey +

zez =

3i=1

xiexi

=

xiexi

. (41)


Position Vecto and Coo dinate S stems Diffe entiation of the Position Vecto K onecke S mbol (K onecke Delta)

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R =R

R(37)

=xex + yey + zezx2 + y2 + z2 . (38)


R

xiexi

= R (39)

= R (40)


=

xex +

yey +

zez =

3i=1

xiexi

=

xiexi

. (41)



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R =R

R(37)

=xex + yey + zezx2 + y2 + z2 . (38)


R

xiexi

= R (39)

= R (40)


=

xex +

yey +

zez =

3i=1

xiexi

=

xiexi

. (41)



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R =R

R(37)

=xex + yey + zezx2 + y2 + z2 . (38)


R

xiexi

= R (39)

= R (40)


=

xex +

yey +

zez =

3i=1

xiexi

=

xiexi

. (41)



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R =R

R(37)

=xex + yey + zezx2 + y2 + z2 . (38)


R

xiexi

= R (39)

= R (40)


=

xex +

yey +

zez =

3i=1

xiexi

=

xiexi

. (41)



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y ; y ( )

Cartesian Coordinate SystemExample: Proof ofR = R

Example (Proof ofR = R)

R

xiexi

=3

i=1

R

xiexi

(42)

=R

x1ex1

+R

x2ex2

+R

x3ex3

(43)

= x1

x21 + x

22 + x

23 ex1

+ x2

x21 + x

22 + x

23 ex2

+ x3

x21 + x

22 + x

23 ex3

.

(44)

We find by applying the chain rule

x1x21 + x22 + x23 = x1 x21 + x22 + x23

1

2

=

1

2x21 + x22 + x23 12

Derivative ofouter function

2 x1Derivative ofinner function

(45)

=x1

x21 + x22 + x231

2

=x1

x21 + x22 + x23. (46)



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y y ( )



R

xiexi

=3

i=1

R

xiexi

(42)

=R

x1ex1

+R

x2ex2

+R

x3ex3

(43)

= x1

x21 + x

22 + x

23 ex1

+ x2

x21 + x

22 + x

23 ex2

+ x3

x21 + x

22 + x

23 ex3

.

(44)


x1x21 + x22 + x23 = x1 x21 + x22 + x23

1

2

=

1

2x21 + x22 + x23 12



(45)

=x1

x21 + x22 + x231

2

=x1

x21 + x22 + x23. (46)



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R

xiexi

=3

i=1

R

xiexi

(42)

=R

x1ex1

+R

x2ex2

+R

x3ex3

(43)

= x1

x21 + x

22 + x

23 ex1

+ x2

x21 + x

22 + x

23 ex2

+ x3

x21 + x

22 + x

23 ex3

.

(44)


x1x21 + x22 + x23 = x1 x21 + x22 + x23

1

2

=

1

2x21 + x22 + x23 12



(45)

=x1

x21 + x22 + x231

2

=x1

x21 + x22 + x23. (46)



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R

xiexi

=3

i=1

R

xiexi

(42)

=R

x1ex1

+R

x2ex2

+R

x3ex3

(43)

= x1

x21 + x

22 + x

23 ex1

+ x2

x21 + x

22 + x

23 ex2

+ x3

x21 + x

22 + x

23 ex3

.

(44)


x1x21 + x22 + x23 = x1 x21 + x22 + x23

1

2

=

1

2x21 + x22 + x23 12



(45)

=x1

x21 + x22 + x231

2

=x1

x21 + x22 + x23. (46)



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R

xiexi

=3

i=1

R

xiexi

(42)

=R

x1ex1

+R

x2ex2

+R

x3ex3

(43)

= x1

x21 + x

22 + x

23 ex1

+ x2

x21 + x

22 + x

23 ex2

+ x3

x21 + x

22 + x

23 ex3

.

(44)


x1x21 + x22 + x23 = x1 x21 + x22 + x23

1

2

=

1

2x21 + x22 + x23 12



(45)

=x1

x21 + x22 + x231

2

=x1

x21 + x22 + x23. (46)



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R

xiexi

=3

i=1

R

xiexi

(42)

=R

x1ex1

+R

x2ex2

+R

x3ex3

(43)

= x1

x21 + x

22 + x

23 ex1

+ x2

x21 + x

22 + x

23 ex2

+ x3

x21 + x

22 + x

23 ex3

.

(44)


x1x21 + x22 + x23 = x1 x21 + x22 + x23

1

2

=

1

2 x21 + x22 + x23 12 Derivative ofouter function


(45)

=x1

x21 + x22 + x231

2

=x1

x21 + x22 + x23. (46)


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R

xiexi

=3

i=1

R

xiexi

(42)

=R

x1ex1

+R

x2ex2

+R

x3ex3

(43)

= x1

x21 + x

22 + x

23 ex1

+ x2

x21 + x

22 + x

23 ex2

+ x3

x21 + x

22 + x

23 ex3

.

(44)


x1x21 + x22 + x23 = x1 x21 + x22 + x23

1

2

=

1

2 x21 + x22 + x23 12 Derivative ofouter function


(45)

=x1

x21 + x22 + x231

2

=x1

x21 + x22 + x23. (46)



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Analog follows for the other two deriviatives

x2x21 + x

22 + x

23 =

x2

x21 + x22 + x23(47)

x3

x21 + x

22 + x

23 =

x3x21 + x

22 + x

23

(48)

or in general

xi

x21 + x

22 + x

23 = x

ix21 + x

22 + x

23

i = 1, 2, 3 . (49)


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Analog follows for the other two deriviatives

x2x21 + x

22 + x

23 =

x2

x21 + x22 + x23(47)

x3

x21 + x

22 + x

23 =

x3x21 + x

22 + x

23

(48)

or in general

xi

x21 + x22 + x23 = xix21 + x

22 + x

23

i = 1, 2, 3 . (49)



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We summarize:

R

xiexi

=

x1x21 + x

22 + x

23 ex1

+

x2x21 + x

22 + x

23 ex2

+

x3x21 + x

22 + x

23 ex3

(50)

=x1

x21 + x22 + x

23

ex1+

x2x21 + x

22 + x

23

ex2+

x3x21 + x

22 + x

23

ex3(51)

=1

x21 + x22 + x

23

x1ex1 + x2ex2 + x3ex3

(52)

=x1ex1

+ x2ex2

+ x3ex3

x21 + x22 + x

23

(53)

=R

R(54)

= R . (55)



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We summarize:

R

xiexi

=

x1x21 + x

22 + x

23 ex1

+

x2x21 + x

22 + x

23 ex2

+

x3x21 + x

22 + x

23 ex3

(50)

= x1

x21 + x22 + x

23

ex1+ x

2x21 + x

22 + x

23

ex2+ x

3x21 + x

22 + x

23

ex3(51)

=1

x21 + x22 + x

23


(52)

=x1ex1

+ x2ex2

+ x3ex3

x21 + x22 + x

23

(53)

=R

R(54)

= R . (55)



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We summarize:

R

xiexi

=

x1x21 + x

22 + x

23 ex1

+

x2x21 + x

22 + x

23 ex2

+

x3x21 + x

22 + x

23 ex3

(50)

= x1

x21 + x22 + x

23

ex1+ x

2x21 + x

22 + x

23

ex2+ x

3x21 + x

22 + x

23

ex3(51)

=1

x21 + x22 + x

23


(52)

=x1ex1

+ x2ex2

+ x3ex3

x21 + x22 + x

23

(53)

=R

R(54)

= R . (55)



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We summarize:

R

xiexi

=

x1x21 + x

22 + x

23 ex1

+

x2x21 + x

22 + x

23 ex2

+

x3x21 + x

22 + x

23 ex3

(50)

= x1

x21 + x22 + x

23

ex1+ x

2x21 + x

22 + x

23

ex2+ x

3x21 + x

22 + x

23

ex3(51)

=1

x21 + x22 + x

23


(52)

=x1e

x1+ x2e

x2+ x3e

x3x21 + x

22 + x

23

(53)

=R

R(54)

= R . (55)



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We summarize:

R

xiexi

=

x1x21 + x

22 + x

23 ex1

+

x2x21 + x

22 + x

23 ex2

+

x3x21 + x

22 + x

23 ex3

(50)

= x1

x21 + x22 + x

23

ex1+ x

2x21 + x

22 + x

23

ex2+ x

3x21 + x

22 + x

23

ex3(51)

=1

x21 + x22 + x

23


(52)

=x1e

x1+ x2e

x2+ x3e

x3x21 + x

22 + x

23

(53)

=R

R(54)

= R . (55)



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We summarize:

R

xiexi

=

x1x21 + x

22 + x

23 ex1

+

x2x21 + x

22 + x

23 ex2

+

x3x21 + x

22 + x

23 ex3

(50)

= x1

x21 + x22 + x

23

ex1+ x

2x21 + x

22 + x

23

ex2+ x

3x21 + x

22 + x

23

ex3(51)

=1

x21 + x22 + x

23


(52)

=x1e

x1+ x2e

x2+ x3e

x3x21 + x

22 + x

23

(53)

=R

R(54)

= R . (55)



C i C di S

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Cartesian Coordinate SystemProjection of the Position Vector in the Direction of the Axes of the Cartesian Coordinate System

Projection of the Position Vector in the Direction ofthe Axes of the Cartesian Coordinate System

If we project the position vector in the direction ofthe Axes of the Cartesian Coordinates System wefind the components ofR:

x = R ex (56)

y = R ey (57)

z = R ez , (58)

this means also

R = (R ex)ex + (R ey)ey + (R ez)ez (59)

Cartesian Coordinates of the spatialpoint P and the related positionvector

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O

x

y

z

z

x

y

P (x,y,z)

R

ez

ex

ey


Position Vector and Coordinate Systems Cylinder Coordinate System

(Ci l )C li d C di S

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(Circular)Cylinder Coordinate SystemCoordinates; Unit Vectors, Magnitude of the Position Vector

Coordinates; Unit Vectors, Magnitude ofthe Position Vector

Cylinder Coordinates: r,,z

in the limits 0 r < , 0 < 2,

< z

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